2532:
2019:
8219:
5224:
5402:
5524:
2527:{\displaystyle {\begin{aligned}D_{0}&=1+{\tfrac {3}{4}}e^{2}+{\tfrac {45}{64}}e^{4}+{\tfrac {175}{256}}e^{6}+{\tfrac {11025}{16384}}e^{8}+\cdots ,\\D_{2}&=-{\tfrac {3}{8}}e^{2}-{\tfrac {15}{32}}e^{4}-{\tfrac {525}{1024}}e^{6}-{\tfrac {2205}{4096}}e^{8}-\cdots ,\\D_{4}&={\tfrac {15}{256}}e^{4}+{\tfrac {105}{1024}}e^{6}+{\tfrac {2205}{16384}}e^{8}+\cdots ,\\D_{6}&=-{\tfrac {35}{3072}}e^{6}-{\tfrac {105}{4096}}e^{8}-\cdots ,\\D_{8}&={\tfrac {315}{131072}}e^{8}+\cdots .\end{aligned}}}
3998:
3188:
4905:
6891:
7339:
1781:
4278:
3625:
2835:
5219:{\displaystyle {\begin{aligned}m(\varphi )&=\left(111\,132.952\,55\,\varphi ^{(\circ )}-16\,038.509\,\sin 2\varphi +16.833\,\sin 4\varphi -0.022\,\sin 6\varphi +0.000\,03\,\sin 8\varphi \right){\mbox{ metres}}\\&=\left(111\,132.952\,55\,\beta ^{(\circ )}-5\,346.170\,\sin 2\beta -1.122\,\sin 4\beta -0.001\,\sin 6\beta -0.5\times 10^{-6}\,\sin 8\beta \right){\mbox{ metres,}}\end{aligned}}}
821:
1186:
6615:
7063:
3993:{\displaystyle {\begin{aligned}B_{0}&=1+{\tfrac {1}{4}}n^{2}+{\tfrac {1}{64}}n^{4}+\cdots =H_{0}\,,\\B_{2}&=-{\tfrac {1}{2}}n+{\tfrac {1}{16}}n^{3}+\cdots ,&B_{6}&=-{\tfrac {1}{48}}n^{3}+\cdots ,\\B_{4}&=-{\tfrac {1}{16}}n^{2}+{\tfrac {1}{64}}n^{4}+\cdots ,\qquad &B_{8}&=-{\tfrac {5}{512}}n^{4}+\cdots .\end{aligned}}}
4013:
1503:
3183:{\displaystyle {\begin{aligned}H_{0}&=1+{\tfrac {1}{4}}n^{2}+{\tfrac {1}{64}}n^{4}+\cdots ,\\H_{2}&=-{\tfrac {3}{2}}n+{\tfrac {3}{16}}n^{3}+\cdots ,&H_{6}&=-{\tfrac {35}{48}}n^{3}+\cdots ,\\H_{4}&={\tfrac {15}{16}}n^{2}-{\tfrac {15}{64}}n^{4}-\cdots ,\qquad &H_{8}&={\tfrac {315}{512}}n^{4}-\cdots .\end{aligned}}}
2008:
2824:
4288:
The above series, to eighth order in eccentricity or fourth order in third flattening, provide millimetre accuracy. With the aid of symbolic algebra systems, they can easily be extended to sixth order in the third flattening which provides full double precision accuracy for terrestrial applications.
5948:
478:
was defined as the length of 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
3614:
6066:
559:
993:
6886:{\displaystyle {\begin{aligned}H'_{2}&={\tfrac {3}{2}}n-{\tfrac {27}{32}}n^{3}+\cdots ,&H'_{6}&={\tfrac {151}{96}}n^{3}+\cdots ,\\H'_{4}&={\tfrac {21}{16}}n^{2}-{\tfrac {55}{32}}n^{4}+\cdots ,\qquad &H'_{8}&={\tfrac {1097}{512}}n^{4}+\cdots .\end{aligned}}}
7334:{\displaystyle {\begin{aligned}B'_{2}&={\tfrac {1}{2}}n-{\tfrac {9}{32}}n^{3}+\cdots ,&B'_{6}&={\tfrac {29}{96}}n^{3}-\cdots ,\\B'_{4}&={\tfrac {5}{16}}n^{2}-{\tfrac {37}{96}}n^{4}+\cdots ,\qquad &B'_{8}&={\tfrac {539}{1536}}n^{4}-\cdots .\end{aligned}}}
321:
over the period 1684–1718. The arc was measured with at least three latitude determinations, so they were able to deduce mean curvatures for the northern and southern halves of the arc, allowing a determination of the overall shape. The results indicated that the Earth was a
4273:{\displaystyle {\begin{aligned}m(\varphi )={\frac {a+b}{2}}{\Biggl (}&B_{0}\varphi -B_{2}\sin 2\varphi +B_{4}\sin 4\varphi -B_{6}\sin 6\varphi +B_{8}\sin 8\varphi -\cdots \\&\quad -{\frac {2n\sin 2\varphi }{\sqrt {1+2n\cos 2\varphi +n^{2}}}}{\Biggr )}.\end{aligned}}}
1776:{\displaystyle {\begin{aligned}m(\varphi )&=a\left(E(\varphi ,e)-{\frac {e^{2}\sin \varphi \cos \varphi }{\sqrt {1-e^{2}\sin {}^{\!2}\varphi }}}\right)\\&=a\left(E(\varphi ,e)+{\frac {d^{2}}{d\varphi ^{2}}}E(\varphi ,e)\right)\\&=bE(\beta ,ie')\,.\end{aligned}}}
4422:
6604:
7052:
405:). It was divided into five parts by four intermediate determinations of latitude. By combining the measurements together with those for the arc of Peru, ellipsoid shape parameters were determined and the distance between the Equator and pole along the
956:
4610:
1298:
6354:
1831:
3391:
2646:
453:
In the 19th century, many astronomers and geodesists were engaged in detailed studies of the Earth's curvature along different meridian arcs. The analyses resulted in a great many model ellipsoids such as
Plessis 1817, Airy 1830,
2024:
5751:
1795:
The above integral may be expressed as an infinite truncated series by expanding the integrand in a Taylor series, performing the resulting integrals term by term, and expressing the result as a trigonometric series. In 1755,
1786:
The calculation (to arbitrary precision) of the elliptic integrals and approximations are also discussed in the NIST handbook. These functions are also implemented in computer algebra programs such as
Mathematica and Maxima.
5446:, is a useful starting point for translations, but translators must revise errors as necessary and confirm that the translation is accurate, rather than simply copy-pasting machine-translated text into the English Knowledge.
4845:
3437:
1484:
816:{\displaystyle {\begin{aligned}f&={\frac {a-b}{a}}\,,\qquad e^{2}=f(2-f)\,,\qquad n={\frac {a-b}{a+b}}={\frac {f}{2-f}}\,,\\b&=a(1-f)=a{\sqrt {1-e^{2}}}\,,\qquad e^{2}={\frac {4n}{(1+n)^{2}}}\,.\end{aligned}}}
5977:
5386:
1181:{\displaystyle {\begin{aligned}m(\varphi )&=\int _{0}^{\varphi }M(\varphi )\,d\varphi \\&=a(1-e^{2})\int _{0}^{\varphi }\left(1-e^{2}\sin ^{2}\varphi \right)^{-{\frac {3}{2}}}\,d\varphi \,.\end{aligned}}}
6425:
5601:
4018:
255:
of gravity was less at
Cayenne than at Paris. Pendulum gravimeters began to be taken on voyages to remote parts of the world, and it was slowly discovered that gravity increases smoothly with increasing
4298:
1348:
Even though latitude is normally confined to the range , all the formulae given here apply to measuring distance around the complete meridian ellipse (including the anti-meridian). Thus the ranges of
5702:
4743:
7068:
6620:
5982:
4910:
3630:
2840:
1508:
998:
564:
4867:. This method avoids the calculation of most of the trigonometric functions and allows the series to be summed rapidly and accurately. The technique can also be used to evaluate the difference
487:: "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".
2627:
4292:
Delambre and Bessel both wrote their series in a form that allows them to be generalized to arbitrary order. The coefficients in Bessel's series can expressed particularly simply
6212:
139:, with an error on the real value between -2.4% and +0.8% (assuming a value for the stadion between 155 and 160 metres). Eratosthenes described his technique in a book entitled
6465:
6910:
5432:
841:
6119:
4433:
5456:
5736:
382:). The resulting measurements at equatorial and polar latitudes confirmed that the Earth was best modelled by an oblate spheroid, supporting Newton. However, by 1743,
66:
in the region of the measurements. Measurements of meridian arcs at several latitudes along many meridians around the world can be combined in order to approximate a
2003:{\displaystyle m(\varphi )={\frac {b^{2}}{a}}\left(D_{0}\varphi +D_{2}\sin 2\varphi +D_{4}\sin 4\varphi +D_{6}\sin 6\varphi +D_{8}\sin 8\varphi +\cdots \right)\,,}
7895:, Druck und Verlag von B. G. Teubner, Leipzig, § 1.7, pp. 44–48. English translation (by the Aeronautical Chart and Information Center, St. Louis) available at
1201:
6254:
2819:{\displaystyle m(\varphi )={\frac {a+b}{2}}\left(H_{0}\varphi +H_{2}\sin 2\varphi +H_{4}\sin 4\varphi +H_{6}\sin 6\varphi +H_{8}\sin 8\varphi +\cdots \right)\,,}
3261:
5943:{\displaystyle m_{\mathrm {p} }={\frac {\pi (a+b)}{4}}c_{0}={\frac {\pi (a+b)}{4}}\sum _{j=0}^{\infty }\left({\frac {(2j-3)!!}{(2j)!!}}\right)^{2}n^{2j}\,,}
7347:
showed that the distance along a geodesic on a spheroid is the same as the distance along the perimeter of an ellipse. For this reason, the expression for
7548:
United States
National Museum Bulletin 240: Contributions from the Museum of History and Technology reprinted in Bulletin of the Smithsonian Institution
7661:
Comparisons of the standards of length of
England, France, Belgium, Prussia, Russia, India, Australia, made at the Ordnance survey office, Southampton
3609:{\displaystyle m(\varphi )={\frac {a+b}{2}}\left(B_{0}\beta +B_{2}\sin 2\beta +B_{4}\sin 4\beta +B_{6}\sin 6\beta +B_{8}\sin 8\beta +\cdots \right),}
5466:
Do not translate text that appears unreliable or low-quality. If possible, verify the text with references provided in the foreign-language article.
4770:
1392:
7735:
Section 5.6. This reference includes the derivation of curvature formulae from first principles and a proof of
Meusnier's theorem. (Supplements:
81:
in the region the survey was to be conducted, leading to a proliferation of reference ellipsoids around the world. The latest determinations use
6061:{\displaystyle {\begin{aligned}m_{\mathrm {p} }&=0.9983242984312529\ {\frac {\pi }{2}}\ a\\&=10\,001\,965.729{\mbox{ m.}}\end{aligned}}}
8056:
7375:, the arc length on the auxiliary sphere. The requisite series extended to sixth order are given by Charles Karney, Eqs. (17) & (21), with
4003:
Because this series provides an expansion for the elliptic integral of the second kind, it can be used to write the arc length in terms of the
281:
7852:"Bestimmung der Axen des elliptischen Rotationssphäroids, welches den vorhandenen Messungen von Meridianbögen der Erde am meisten entspricht"
5622:
5327:
6378:
5549:
8030:
1490:
1375:
4417:{\displaystyle B_{2k}={\begin{cases}c_{0}\,,&{\text{if }}k=0\,,\\{\dfrac {c_{k}}{k}}\,,&{\text{if }}k>0\,,\end{cases}}}
8027:
A General
Formula for Calculating Meridian Arc Length and its Application to Coordinate Conversion in the Gauss-Krüger Projection
3405:
402:
331:
7499:
7938:
4643:
7693:
5451:
7645:
7422:
5474:
355:
89:
to determine reference ellipsoids, especially the geocentric ellipsoids now used for global coordinate systems such as
7659:
2562:
7633:
5631:
5487:
Content in this edit is translated from the existing German
Knowledge article at ]; see its history for attribution.
3409:
8263:
5739:
1801:
8258:
3401:
3400:. Even though this results in more slowly converging series, such series are used in the specification for the
520:
367:
7637:
7551:
4864:
1816:
463:
390:
343:
4323:
115:
Early estimations of Earth's size are recorded from Greece in the 4th century BC, and from scholars at the
6154:
2550:
8047:
6599:{\displaystyle \varphi =\mu +H'_{2}\sin 2\mu +H'_{4}\sin 4\mu +H'_{6}\sin 6\mu +H'_{8}\sin 8\mu +\cdots }
5965:
7765:
7047:{\displaystyle \beta =\mu +B'_{2}\sin 2\mu +B'_{4}\sin 4\mu +B'_{6}\sin 6\mu +B'_{8}\sin 8\mu +\cdots ,}
951:{\displaystyle M(\varphi )={\frac {a(1-e^{2})}{\left(1-e^{2}\sin ^{2}\varphi \right)^{\frac {3}{2}}}},}
539:
480:
327:
261:
8103:
7927:
7442:
7356:
4605:{\displaystyle c_{k}=\sum _{j=0}^{\infty }{\frac {(2j-3)!!\,(2j+2k-3)!!}{(2j)!!\,(2j+2k)!!}}n^{k+2j}}
3428:
505:. On an ellipsoid of revolution, for short meridian arcs, their length can be approximated using the
371:
314:
8218:
7926:
J. W. Hager, J.F. Behensky, and B.W. Drew, 1989. Defense
Mapping Agency Technical Report TM 8358.2.
1192:
7474:
6072:
5482:
4749:
2637:
7806:
7543:
6081:
509:
and the circular arc formulation. For longer arcs, the length follows from the subtraction of two
8238:
7427:
6145:
5607:
359:
351:
7928:
The universal grids: Universal
Transverse Mercator (UTM) and Universal Polar Stereographic (UPS)
164:
8051:
7793:
8127:
7889:
7794:"Élémens de la trigonométrie sphéroïdique tirés de la méthode des plus grands et plus petits"
7601:
7344:
5503:
4896:
459:
383:
8141:
7854:[Estimation of the axes of the ellipsoid through measurements of the meridian arc].
7572:
7489:
7466:
1293:{\displaystyle m(\varphi )=b\int _{0}^{\beta }{\sqrt {1+e'^{2}\sin ^{2}\beta }}\,d\beta \,,}
502:
213:, by the 17th century, evidence was accumulating that it was not a perfect sphere. In 1672,
8191:
7978:
7863:
7822:
7625:
7578:, translated into English by Andrew Motte. A searchable modern translation is available at
7528:
6349:{\displaystyle \varphi _{i+1}=\varphi _{i}-{\frac {m(\varphi _{i})-m}{M(\varphi _{i})}}\,,}
4847:
appearing in the above formula and results in poorer convergence of the series in terms of
326:
spheroid (with an equatorial radius less than the polar radius). To resolve the issue, the
110:
77:
required a single arc. Accurate survey work beginning in the 19th century required several
44:
7796:[Elements of spheroidal trigonometry taken from the method of maxima and minima].
6128:
of a meridian ellipse can also be rewritten in the form of a rectifying circle perimeter,
5710:
553:. Only two of these parameters are independent and there are many relations between them:
8:
7467:
7437:
7432:
7411:
6431:
3421:
3386:{\displaystyle {\tfrac {1}{2}}(a+b)={\frac {a}{1+n}}=a(1-n+n^{2}-n^{3}+n^{4}-\cdots )\,,}
2549:
Series with considerably faster convergence can be obtained by expanding in terms of the
206:
59:
55:
8195:
7982:
7867:
7583:
375:
8207:
8181:
8133:
Exercices de Calcul Intégral sur Divers Ordres de Transcendantes et sur les Quadratures
8073:
7994:
7968:
7959:(2010). "The calculation of longitude and latitude from geodesic measurements (1825)".
7401:
6245:
221:
was not constant over the Earth (as it would be if the Earth were a sphere); he took a
106:
82:
7605:
832:
506:
8211:
8145:
8077:
7998:
7689:
7665:
7664:. London: G.E. Eyre and W. Spottiswoode for H.M. Stationery Office. pp. 281–87.
7641:
7629:
7611:
7495:
5496:
5478:
4004:
398:
265:
86:
538:, but in theoretical work it is useful to define extra parameters, particularly the
8253:
8199:
8065:
7986:
7956:
7896:
7871:
7847:
7727:
4623:
2633:
455:
423:
as specified by the standard toise bar in Paris. Defining this distance as exactly
347:
78:
32:
8131:
7522:
6459:
Alternatively, Helmert's series for the meridian distance can be reverted to give
519:. This is an important problem in the theory of map projections, particularly the
58:. One or more measurements of meridian arcs can be used to infer the shape of the
363:
318:
210:
148:
147:
about 150 years later, and slightly better results were calculated in 827 by the
120:
74:
8223:
7913:
496:
8099:
7789:
7621:
7416:
5536:
5518:
5414:
1797:
406:
379:
335:
309:. This was disputed by some, but not all, French scientists. A meridian arc of
222:
136:
40:
8239:
Online computation of meridian arcs on different geodetic reference ellipsoids
8203:
8069:
7752:
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors, 2010,
3420:
In 1825, Bessel derived an expansion of the meridian distance in terms of the
8247:
8149:
7875:
5611:
5523:
4637:
The coefficients in Helmert's series can similarly be expressed generally by
475:
230:
197:
are used interchangeably in this article, with oblate implied if not stated.
8026:
5971:
The numerical expression for the quarter meridian on the WGS84 ellipsoid is
4840:{\displaystyle {\frac {2n\sin 2\varphi }{\sqrt {1+2n\cos 2\varphi +n^{2}}}}}
1479:{\displaystyle m(\varphi )=a\left(1-e^{2}\right)\,\Pi (\varphi ,e^{2},e)\,.}
1369:
7990:
7615:
5615:
983:
in radians). Therefore, the meridian distance from the equator to latitude
276:
252:
152:
132:
7669:
7900:
7731:
7462:
6236:
In some problems, we need to be able to solve the inverse problem: given
4863:
The trigonometric series given above can be conveniently evaluated using
484:
339:
310:
214:
5745:
The quarter meridian is also given by the following generalized series:
4895:
Substituting the values for the semi-major axis and eccentricity of the
3232:
is constant under this interchange, half the terms in the expansions of
185:, although the qualifying words "of revolution" are usually dropped. An
5485:
to the source of your translation. A model attribution edit summary is
547:
290:
189:
that is not an ellipsoid of revolution is called a triaxial ellipsoid.
144:
128:
48:
24:
5381:{\displaystyle \Delta m=(111\,133-560\cos 2\varphi ){\mbox{ metres.}}}
6125:
186:
135:
about 240 BC. He estimated that the meridian has a length of 252,000
36:
7891:
Die mathematischen und physikalischen Theorieen der höheren Geodäsie
7851:
7709:
7355:
and its inverse given above play a key role in the solution of the
6420:{\displaystyle \mu ={\frac {\pi }{2}}{\frac {m}{m_{\mathrm {p} }}}}
5596:{\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,.}
286:
257:
182:
8186:
8108:. US Coast and Geodetic Survey Special Publication No. 67. p. 127.
7973:
7740:
7736:
7406:
6434:. Note that it there is no need to differentiate the series for
6231:
394:
269:
226:
218:
124:
20:
7544:"Paper 44: Development of gravity pendulums in the 19th century"
5443:
127:
in the 9th century. The first realistic value was calculated by
7824:
Méthodes Analytiques pour la Détermination d'un Arc du Méridien
5317:
between the two parallels at ±0.5° from the circle at latitude
2636:
obtained one such series, which was put into a simpler form by
1494:
1383:
174:
116:
90:
961:
The arc length of an infinitesimal element of the meridian is
7826:; précédées d'un mémoire sur le même sujet par A. M. Legendre
3415:
1191:
The distance formula is simpler when written in terms of the
435:
420:
248:
143:, which has not been preserved. A similar method was used by
63:
8105:
Latitude Developments Connected With Geodesy and Cartography
7579:
7520:
7917:. Royal Prussian Geodetic Institute, New Series 52, page 12
4410:
1379:
7541:
6359:
until convergence. A suitable starting guess is given by
5455:
to this template: there are already 1,879 articles in the
4626:, extended to negative values via the recursion relation:
35:
between two points on the Earth's surface having the same
16:
Distance along a portion of a meridian, for use in geodesy
7798:
Mémoires de l'Académie Royale des Sciences de Berlin 1753
7648:), the first chapter covers the history of early surveys.
6445:, since the formula for the meridian radius of curvature
5267:
On the ellipsoid the exact distance between parallels at
2540:
2537:
Richard Rapp gives a detailed derivation of this result.
515:, the distance from the equator to a point at a latitude
177:"squashed at the poles". Modern literature uses the term
54:
The purpose of measuring meridian arcs is to determine a
7776:
7753:
5310:. For WGS84 an approximate expression for the distance
7688:. Hamble: The Royal Yachting Association. p. 76.
7297:
7243:
7218:
7163:
7110:
7092:
6849:
6795:
6770:
6715:
6662:
6644:
6048:
5621:
The quarter meridian can be expressed in terms of the
5372:
5206:
5055:
3956:
3902:
3877:
3822:
3769:
3751:
3682:
3657:
3266:
3146:
3095:
3070:
3018:
2965:
2947:
2892:
2867:
2490:
2438:
2413:
2358:
2333:
2308:
2256:
2231:
2206:
2181:
2126:
2101:
2076:
2051:
1374:
The above integral is related to a special case of an
8172:
Karney, C. F. F. (2013). "Algorithms for geodesics".
7481:
7066:
6913:
6618:
6468:
6381:
6257:
6157:
6084:
5980:
5754:
5713:
5634:
5552:
5330:
4908:
4773:
4646:
4436:
4365:
4301:
4016:
3628:
3440:
3264:
2838:
2649:
2565:
2022:
1834:
1807:
1506:
1395:
1204:
996:
844:
562:
8052:"A new series for the rectification of the ellipsis"
5439:
4738:{\displaystyle H_{2k}=(-1)^{k}(1-2k)(1+2k)B_{2k}\,.}
462:. A comprehensive list of ellipsoids is given under
7521:Poynting, John Henry; Joseph John Thompson (1907).
5435:
a machine-translated version of the German article.
501:On a sphere, the meridian arc length is simply the
94:
7779:, 2009, A computer algebra system, version 5.20.1.
7333:
7046:
6885:
6598:
6419:
6348:
6206:
6113:
6060:
5942:
5730:
5696:
5595:
5380:
5218:
4839:
4737:
4604:
4416:
4272:
3992:
3608:
3385:
3182:
2818:
2621:
2526:
2002:
1775:
1478:
1292:
1180:
950:
815:
7914:Konforme Abbildung des Erdellipsoids in der Ebene
4258:
4056:
3205:are interchanged, and because the initial factor
2556:instead of the eccentricity. They are related by
1618:
1363:
8245:
2622:{\displaystyle e^{2}={\frac {4n}{(1+n)^{2}}}\,.}
1491:incomplete elliptic integrals of the second kind
393:had remeasured and extended the French arc from
5697:{\displaystyle m_{\mathrm {p} }=aE(e)=bE(ie').}
5531:The distance from the equator to the pole, the
3255:as the initial factor by writing, for example,
8057:Transactions of the Royal Society of Edinburgh
7939:A guide to coordinate systems in Great Britain
7636:). In addition the book has been reprinted by
7494:. De Gruyter Textbook. De Gruyter. p. 5.
6232:The inverse meridian problem for the ellipsoid
5481:accompanying your translation by providing an
5426:Click for important translation instructions.
5413:expand this section with text translated from
1376:incomplete elliptic integral of the third kind
7658:Clarke, Alexander Ross; James, Henry (1866).
7542:Victor F., Lenzen; Robert P. Multauf (1964).
5623:complete elliptic integral of the second kind
386:had completely supplanted Newton's approach.
73:The earliest determinations of the size of a
8140:] (in French). Paris: Courcier. p.
7527:. London: Charles Griffin & Co. p.
4767:originates from the additional expansion of
7951:
7949:
7947:
7828:, De L'Imprimerie de Crapelet, Paris, 72–73
7817:
7815:
7657:
7487:
1819:in 1799 derived a widely used expansion on
479:from the latitude scale of charts. As the
7455:
5964:above.) This result was first obtained by
4892:while maintaining high relative accuracy.
3416:Series in terms of the parametric latitude
434:led to the construction of a new standard
200:
8185:
8093:
8031:Geospatial Information Authority of Japan
7972:
7683:
7535:
6342:
6043:
6039:
5936:
5589:
5346:
5187:
5152:
5133:
5114:
5110:
5087:
5083:
5079:
5036:
5032:
5013:
4994:
4975:
4971:
4948:
4944:
4940:
4731:
4552:
4498:
4403:
4383:
4356:
4336:
3722:
3379:
2812:
2615:
1996:
1765:
1472:
1440:
1286:
1279:
1170:
1163:
1047:
805:
751:
689:
631:
595:
100:
8126:
7944:
7812:
7596:
7594:
7592:
7576:, Book III, Proposition XIX, Problem III
5522:
5257:expressed in degrees (and similarly for
4858:
3396:and expanding the result as a series in
3247:The series can be expressed with either
285:as a proof that the Earth was an oblate
247:minutes per day compared to its rate at
7754:NIST Handbook of Mathematical Functions
7721:
7715:
7367:, the distance along the geodesic, and
6075:is simply four times quarter meridian:
5961:
3406:National Geospatial-Intelligence Agency
8246:
8171:
8003:English translation of Astron. Nachr.
7955:
7846:
7600:
507:Earth's meridional radius of curvature
8046:
7788:
7766:Mathematica guide: Elliptic Integrals
7589:
7565:
7461:
5527:A quarter meridian or Earth quadrant.
4283:
526:The main ellipsoidal parameters are,
7932:
6207:{\displaystyle M_{r}=0.5(a+b)/c_{0}}
5614:, and used in the definition of the
5395:
2541:Expansions in the third flattening (
1790:
469:
403:meridian arc of Delambre and Méchain
165:Earth ellipsoid § Determination
158:
7941:, Ordnance Survey of Great Britain.
5391:
1489:It may also be written in terms of
13:
7423:French Geodesic Mission to Lapland
6409:
5991:
5853:
5761:
5641:
5559:
5331:
4466:
1441:
70:intended to fit the entire world.
14:
8275:
8232:
7837:Rapp, R, (1991), §3.6, pp. 36–40.
7419:(West Europe-Africa Meridian-arc)
205:Although it had been known since
151:method, attributed to the Caliph
39:. The term may refer either to a
8217:
7674:Appendix on Figure of the Earth.
5400:
4752:and proved by Kazushige Kawase.
3410:Ordnance Survey of Great Britain
1808:Expansions in the eccentricity (
1378:. In the notation of the online
313:was extended to a longer arc by
264:being about 0.5% greater at the
85:measurements and the methods of
8165:
8156:
8120:
8111:
8084:
8040:
8019:
8010:
7920:
7905:
7882:
7840:
7831:
7782:
7770:
7759:
7746:
7686:RYA day skipper handbook - sail
7273:
6896:Similarly, Bessel's series for
6825:
4748:This result was conjectured by
4189:
3932:
3427:in connection with his work on
3125:
755:
635:
599:
448:
169:Early literature uses the term
8138:Exercises in Integral Calculus
7702:
7677:
7651:
7514:
7488:Torge, W.; Müller, J. (2012).
6336:
6323:
6309:
6296:
6186:
6174:
5901:
5892:
5881:
5866:
5828:
5816:
5788:
5776:
5688:
5674:
5662:
5656:
5606:It was part of the historical
5491:You may also add the template
5368:
5340:
5099:
5093:
4960:
4954:
4922:
4916:
4715:
4700:
4697:
4682:
4673:
4663:
4571:
4553:
4543:
4534:
4523:
4499:
4489:
4474:
4030:
4024:
3450:
3444:
3402:transverse Mercator projection
3376:
3319:
3289:
3277:
2659:
2653:
2603:
2590:
1844:
1838:
1762:
1742:
1718:
1706:
1668:
1656:
1553:
1541:
1520:
1514:
1469:
1444:
1405:
1399:
1364:Relation to elliptic integrals
1356:, and the rectifying latitude
1214:
1208:
1086:
1067:
1044:
1038:
1010:
1004:
885:
866:
854:
848:
835:can be shown to be equal to:
793:
780:
722:
710:
628:
616:
521:transverse Mercator projection
490:
217:found the first evidence that
1:
7756:(Cambridge University Press).
7620:. Freely available online at
7552:Smithsonian Institution Press
7524:A Textbook of Physics, 4th Ed
7449:
5493:{{Translated|de|Erdquadrant}}
826:
391:Jean Baptiste Joseph Delambre
7473:. Berlin: Springer. p.
6114:{\displaystyle C_{p}=4m_{p}}
5519:Ellipse § Circumference
1800:derived an expansion in the
833:meridian radius of curvature
497:Latitude § Meridian arc
7:
7821:Delambre, J. B. J. (1799):
7610:. Oxford: Clarendon Press.
7394:
5463:will aid in categorization.
389:By the end of the century,
141:On the measure of the Earth
62:that best approximates the
10:
8280:
5516:
5438:Machine translation, like
1367:
494:
481:Royal Yachting Association
328:French Academy of Sciences
262:gravitational acceleration
162:
104:
8204:10.1007/s00190-012-0578-z
8070:10.1017/s0080456800030817
7856:Astronomische Nachrichten
7710:Geometric Geodesy, Part I
7443:Geodesics on an ellipsoid
5738:are the first and second
5415:the corresponding article
1370:Ellipse § Arc length
315:Giovanni Domenico Cassini
7876:10.1002/asna.18370142301
7724:The Mercator Projections
7684:Hopkinson, Sara (2012).
7469:The Forgotten Revolution
6904:can be reverted to give
6244:. This may be solved by
8162:Helmert (1880), Chap. 5
7893:, Einleitung und 1 Teil
7888:Helmert, F. R. (1880):
7722:Osborne, Peter (2013),
7582:. Search the following
7428:French Geodesic Mission
6217:It can be evaluated as
6146:rectifying Earth radius
5608:definition of the metre
5502:For more guidance, see
4851:compared to the one in
1493:(See the NIST handbook
483:says in its manual for
233:and found that it lost
201:17th and 18th centuries
179:ellipsoid of revolution
8264:History of measurement
7991:10.1002/asna.201011352
7741:Latex code and figures
7602:Clarke, Alexander Ross
7335:
7048:
6887:
6600:
6421:
6350:
6208:
6115:
6062:
5944:
5857:
5732:
5698:
5597:
5528:
5382:
5220:
4841:
4739:
4606:
4470:
4418:
4274:
3994:
3610:
3387:
3184:
2820:
2623:
2528:
2004:
1777:
1480:
1294:
1182:
952:
817:
251:. This indicated the
101:History of measurement
8259:Meridians (geography)
8090:Helmert (1880), §1.10
8016:Helmert (1880), §1.11
8007:, 241–254 (1825), §5.
7345:Adrien-Marie Legendre
7336:
7049:
6888:
6601:
6456:can be used instead.
6422:
6351:
6209:
6116:
6073:Earth's circumference
6063:
5945:
5837:
5733:
5699:
5598:
5539:), also known as the
5526:
5504:Knowledge:Translation
5475:copyright attribution
5383:
5221:
4859:Numerical expressions
4842:
4740:
4607:
4450:
4419:
4275:
3995:
3611:
3388:
3185:
2821:
2624:
2529:
2005:
1778:
1481:
1368:Further information:
1295:
1183:
953:
818:
279:had published in the
105:Further information:
95:numerical expressions
8117:Helmert (1880), §5.6
7901:10.5281/zenodo.32050
7732:10.5281/zenodo.35392
7712:, §3.5.1, pp. 28–32.
7387:playing the role of
7379:playing the role of
7064:
6911:
6616:
6466:
6379:
6255:
6155:
6082:
5978:
5953:(For the formula of
5752:
5731:{\displaystyle e,e'}
5711:
5632:
5550:
5328:
4906:
4771:
4644:
4434:
4299:
4014:
3626:
3438:
3262:
2836:
2647:
2563:
2020:
1832:
1504:
1393:
1360:, are unrestricted.
1202:
994:
842:
560:
458:, Everest 1830, and
111:History of the metre
68:geocentric ellipsoid
8196:2013JGeod..87...43K
8025:Kawase, K. (2011):
7983:2010AN....331..852K
7911:Krüger, L. (1912):
7868:1837AN.....14..333B
7438:Rectifying latitude
7433:Struve Geodetic Arc
7412:Reference ellipsoid
7288:
7209:
7154:
7083:
7022:
6994:
6966:
6938:
6840:
6761:
6706:
6635:
6577:
6549:
6521:
6493:
6432:rectifying latitude
5962:#Generalized series
3422:parametric latitude
1237:
1193:parametric latitude
1103:
1034:
503:circular arc length
332:expeditions to Peru
209:that the Earth was
207:classical antiquity
60:reference ellipsoid
56:figure of the Earth
8174:Journal of Geodesy
8029:, Bulletin of the
7402:History of geodesy
7331:
7329:
7306:
7276:
7252:
7227:
7197:
7172:
7142:
7119:
7101:
7071:
7044:
7010:
6982:
6954:
6926:
6883:
6881:
6858:
6828:
6804:
6779:
6749:
6724:
6694:
6671:
6653:
6623:
6596:
6565:
6537:
6509:
6481:
6417:
6346:
6204:
6111:
6058:
6056:
6052:
6005:0.9983242984312529
5940:
5728:
5694:
5593:
5535:(analogous to the
5529:
5483:interlanguage link
5378:
5376:
5216:
5214:
5210:
5059:
4865:Clenshaw summation
4837:
4735:
4602:
4414:
4409:
4381:
4284:Generalized series
4270:
4268:
3990:
3988:
3965:
3911:
3886:
3831:
3778:
3760:
3691:
3666:
3606:
3383:
3275:
3197:changes sign when
3180:
3178:
3155:
3104:
3079:
3027:
2974:
2956:
2901:
2876:
2816:
2619:
2524:
2522:
2499:
2447:
2422:
2367:
2342:
2317:
2265:
2240:
2215:
2190:
2135:
2110:
2085:
2060:
2000:
1802:third eccentricity
1773:
1771:
1476:
1290:
1223:
1178:
1176:
1089:
1020:
948:
813:
811:
512:meridian distances
409:was calculated as
384:Clairaut's theorem
266:geographical poles
107:History of geodesy
7708:Rapp, R, (1991):
7501:978-3-11-025000-8
7305:
7251:
7226:
7171:
7118:
7100:
6857:
6803:
6778:
6723:
6670:
6652:
6415:
6396:
6340:
6144:. Therefore, the
6051:
6022:
6018:
6009:
5911:
5835:
5795:
5583:
5515:
5514:
5427:
5423:
5375:
5209:
5058:
4835:
4834:
4755:The extra factor
4750:Friedrich Helmert
4581:
4392:
4380:
4345:
4254:
4253:
4052:
4005:geodetic latitude
3964:
3910:
3885:
3830:
3777:
3759:
3690:
3665:
3472:
3311:
3274:
3154:
3103:
3078:
3026:
2973:
2955:
2900:
2875:
2681:
2613:
2498:
2446:
2421:
2366:
2341:
2316:
2264:
2239:
2214:
2189:
2134:
2109:
2084:
2059:
1865:
1791:Series expansions
1701:
1628:
1627:
1277:
1159:
943:
940:
803:
749:
687:
666:
593:
470:The nautical mile
399:Mediterranean Sea
330:(1735) undertook
159:Ellipsoidal Earth
87:satellite geodesy
8271:
8227:
8222:
8221:
8215:
8189:
8169:
8163:
8160:
8154:
8153:
8124:
8118:
8115:
8109:
8097:
8091:
8088:
8082:
8081:
8044:
8038:
8023:
8017:
8014:
8008:
8002:
7976:
7953:
7942:
7936:
7930:
7924:
7918:
7909:
7903:
7886:
7880:
7879:
7862:(333): 333–346.
7844:
7838:
7835:
7829:
7819:
7810:
7805:
7786:
7780:
7774:
7768:
7763:
7757:
7750:
7744:
7734:
7719:
7713:
7706:
7700:
7699:
7681:
7675:
7673:
7655:
7649:
7619:
7598:
7587:
7569:
7563:
7562:
7560:
7559:
7539:
7533:
7532:
7518:
7512:
7511:
7509:
7508:
7485:
7479:
7478:
7472:
7459:
7390:
7386:
7382:
7378:
7374:
7370:
7366:
7362:
7357:geodesic problem
7354:
7350:
7340:
7338:
7337:
7332:
7330:
7317:
7316:
7307:
7298:
7284:
7263:
7262:
7253:
7244:
7238:
7237:
7228:
7219:
7205:
7183:
7182:
7173:
7164:
7150:
7130:
7129:
7120:
7111:
7102:
7093:
7079:
7053:
7051:
7050:
7045:
7018:
6990:
6962:
6934:
6903:
6899:
6892:
6890:
6889:
6884:
6882:
6869:
6868:
6859:
6850:
6836:
6815:
6814:
6805:
6796:
6790:
6789:
6780:
6771:
6757:
6735:
6734:
6725:
6716:
6702:
6682:
6681:
6672:
6663:
6654:
6645:
6631:
6605:
6603:
6602:
6597:
6573:
6545:
6517:
6489:
6455:
6444:
6426:
6424:
6423:
6418:
6416:
6414:
6413:
6412:
6399:
6397:
6389:
6371:
6355:
6353:
6352:
6347:
6341:
6339:
6335:
6334:
6318:
6308:
6307:
6291:
6286:
6285:
6273:
6272:
6243:
6239:
6227:
6225:
6222:
6213:
6211:
6210:
6205:
6203:
6202:
6193:
6167:
6166:
6143:
6120:
6118:
6117:
6112:
6110:
6109:
6094:
6093:
6067:
6065:
6064:
6059:
6057:
6053:
6049:
6029:
6020:
6019:
6011:
6007:
5996:
5995:
5994:
5949:
5947:
5946:
5941:
5935:
5934:
5922:
5921:
5916:
5912:
5910:
5890:
5864:
5856:
5851:
5836:
5831:
5811:
5806:
5805:
5796:
5791:
5771:
5766:
5765:
5764:
5737:
5735:
5734:
5729:
5727:
5703:
5701:
5700:
5695:
5687:
5646:
5645:
5644:
5602:
5600:
5599:
5594:
5588:
5584:
5576:
5564:
5563:
5562:
5533:quarter meridian
5494:
5488:
5462:
5461:|topic=
5459:, and specifying
5444:Google Translate
5425:
5421:
5404:
5403:
5396:
5392:Quarter meridian
5387:
5385:
5384:
5379:
5377:
5373:
5320:
5316:
5309:
5284:
5275:
5263:
5256:
5252:
5251:
5249:
5248:
5245:
5242:
5225:
5223:
5222:
5217:
5215:
5211:
5207:
5204:
5200:
5186:
5185:
5103:
5102:
5064:
5060:
5056:
5053:
5049:
4964:
4963:
4899:ellipsoid gives
4891:
4854:
4850:
4846:
4844:
4843:
4838:
4836:
4833:
4832:
4796:
4795:
4775:
4766:
4744:
4742:
4741:
4736:
4730:
4729:
4681:
4680:
4659:
4658:
4633:
4629:
4624:double factorial
4621:
4611:
4609:
4608:
4603:
4601:
4600:
4582:
4580:
4532:
4472:
4469:
4464:
4446:
4445:
4423:
4421:
4420:
4415:
4413:
4412:
4393:
4390:
4382:
4376:
4375:
4366:
4346:
4343:
4335:
4334:
4314:
4313:
4279:
4277:
4276:
4271:
4269:
4262:
4261:
4255:
4252:
4251:
4215:
4214:
4194:
4185:
4163:
4162:
4138:
4137:
4113:
4112:
4088:
4087:
4072:
4071:
4060:
4059:
4053:
4048:
4037:
3999:
3997:
3996:
3991:
3989:
3976:
3975:
3966:
3957:
3944:
3943:
3922:
3921:
3912:
3903:
3897:
3896:
3887:
3878:
3865:
3864:
3842:
3841:
3832:
3823:
3810:
3809:
3789:
3788:
3779:
3770:
3761:
3752:
3739:
3738:
3721:
3720:
3702:
3701:
3692:
3683:
3677:
3676:
3667:
3658:
3642:
3641:
3615:
3613:
3612:
3607:
3602:
3598:
3579:
3578:
3554:
3553:
3529:
3528:
3504:
3503:
3488:
3487:
3473:
3468:
3457:
3426:
3399:
3392:
3390:
3389:
3384:
3369:
3368:
3356:
3355:
3343:
3342:
3312:
3310:
3296:
3276:
3267:
3254:
3250:
3243:
3231:
3221:
3219:
3218:
3215:
3212:
3204:
3200:
3196:
3189:
3187:
3186:
3181:
3179:
3166:
3165:
3156:
3147:
3137:
3136:
3115:
3114:
3105:
3096:
3090:
3089:
3080:
3071:
3061:
3060:
3038:
3037:
3028:
3019:
3006:
3005:
2985:
2984:
2975:
2966:
2957:
2948:
2935:
2934:
2912:
2911:
2902:
2893:
2887:
2886:
2877:
2868:
2852:
2851:
2825:
2823:
2822:
2817:
2811:
2807:
2788:
2787:
2763:
2762:
2738:
2737:
2713:
2712:
2697:
2696:
2682:
2677:
2666:
2634:Friedrich Bessel
2628:
2626:
2625:
2620:
2614:
2612:
2611:
2610:
2588:
2580:
2575:
2574:
2555:
2551:third flattening
2544:
2533:
2531:
2530:
2525:
2523:
2510:
2509:
2500:
2491:
2481:
2480:
2458:
2457:
2448:
2439:
2433:
2432:
2423:
2414:
2401:
2400:
2378:
2377:
2368:
2359:
2353:
2352:
2343:
2334:
2328:
2327:
2318:
2309:
2299:
2298:
2276:
2275:
2266:
2257:
2251:
2250:
2241:
2232:
2226:
2225:
2216:
2207:
2201:
2200:
2191:
2182:
2169:
2168:
2146:
2145:
2136:
2127:
2121:
2120:
2111:
2102:
2096:
2095:
2086:
2077:
2071:
2070:
2061:
2052:
2036:
2035:
2009:
2007:
2006:
2001:
1995:
1991:
1972:
1971:
1947:
1946:
1922:
1921:
1897:
1896:
1881:
1880:
1866:
1861:
1860:
1851:
1824:
1811:
1782:
1780:
1779:
1774:
1772:
1761:
1729:
1725:
1721:
1702:
1700:
1699:
1698:
1685:
1684:
1675:
1638:
1634:
1630:
1629:
1623:
1622:
1616:
1607:
1606:
1591:
1590:
1571:
1570:
1560:
1495:Section 19.6(iv)
1485:
1483:
1482:
1477:
1462:
1461:
1439:
1435:
1434:
1433:
1384:Section 19.2(ii)
1359:
1355:
1351:
1344:
1343:
1341:
1340:
1334:
1331:
1317:
1299:
1297:
1296:
1291:
1278:
1270:
1269:
1260:
1259:
1258:
1239:
1236:
1231:
1187:
1185:
1184:
1179:
1177:
1162:
1161:
1160:
1152:
1146:
1142:
1135:
1134:
1125:
1124:
1102:
1097:
1085:
1084:
1057:
1033:
1028:
986:
982:
978:
957:
955:
954:
949:
944:
942:
941:
933:
931:
927:
920:
919:
910:
909:
888:
884:
883:
861:
822:
820:
819:
814:
812:
804:
802:
801:
800:
778:
770:
765:
764:
750:
748:
747:
732:
688:
686:
672:
667:
665:
654:
643:
609:
608:
594:
589:
578:
552:
546:, and the third
545:
537:
533:
529:
518:
444:
443:
433:
431:
428:
418:
417:
414:
348:Antonio de Ulloa
308:
306:
305:
302:
299:
246:
245:
241:
238:
79:arc measurements
8279:
8278:
8274:
8273:
8272:
8270:
8269:
8268:
8244:
8243:
8235:
8230:
8216:
8170:
8166:
8161:
8157:
8128:Legendre, A. M.
8125:
8121:
8116:
8112:
8098:
8094:
8089:
8085:
8045:
8041:
8024:
8020:
8015:
8011:
7954:
7945:
7937:
7933:
7925:
7921:
7910:
7906:
7887:
7883:
7845:
7841:
7836:
7832:
7820:
7813:
7787:
7783:
7775:
7771:
7764:
7760:
7751:
7747:
7720:
7716:
7707:
7703:
7696:
7695:9781-9051-04949
7682:
7678:
7656:
7652:
7626:Forgotten Books
7599:
7590:
7586:for 'spheroid'.
7570:
7566:
7557:
7555:
7540:
7536:
7519:
7515:
7506:
7504:
7502:
7486:
7482:
7460:
7456:
7452:
7447:
7397:
7388:
7384:
7380:
7376:
7372:
7368:
7364:
7360:
7352:
7348:
7328:
7327:
7312:
7308:
7296:
7289:
7280:
7274:
7258:
7254:
7242:
7233:
7229:
7217:
7210:
7201:
7194:
7193:
7178:
7174:
7162:
7155:
7146:
7140:
7125:
7121:
7109:
7091:
7084:
7075:
7067:
7065:
7062:
7061:
7014:
6986:
6958:
6930:
6912:
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6908:
6901:
6897:
6880:
6879:
6864:
6860:
6848:
6841:
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6810:
6806:
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6769:
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6753:
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6707:
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6677:
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6643:
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6627:
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6485:
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6380:
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6319:
6303:
6299:
6292:
6290:
6281:
6277:
6262:
6258:
6256:
6253:
6252:
6246:Newton's method
6241:
6237:
6234:
6223:
6220:
6218:
6198:
6194:
6189:
6162:
6158:
6156:
6153:
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6105:
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6089:
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5708:
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5575:
5571:
5558:
5557:
5553:
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5521:
5511:
5510:
5509:
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5460:
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5394:
5371:
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3248:
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3110:
3106:
3094:
3085:
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3069:
3062:
3056:
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3049:
3048:
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3017:
3007:
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2606:
2602:
2589:
2581:
2579:
2570:
2566:
2564:
2561:
2560:
2553:
2547:
2542:
2521:
2520:
2505:
2501:
2489:
2482:
2476:
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2437:
2428:
2424:
2412:
2402:
2396:
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2373:
2369:
2357:
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2332:
2323:
2319:
2307:
2300:
2294:
2290:
2287:
2286:
2271:
2267:
2255:
2246:
2242:
2230:
2221:
2217:
2205:
2196:
2192:
2180:
2170:
2164:
2160:
2157:
2156:
2141:
2137:
2125:
2116:
2112:
2100:
2091:
2087:
2075:
2066:
2062:
2050:
2037:
2031:
2027:
2023:
2021:
2018:
2017:
1967:
1963:
1942:
1938:
1917:
1913:
1892:
1888:
1876:
1872:
1871:
1867:
1856:
1852:
1850:
1833:
1830:
1829:
1820:
1814:
1809:
1793:
1770:
1769:
1754:
1727:
1726:
1694:
1690:
1686:
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1674:
1652:
1648:
1636:
1635:
1617:
1615:
1614:
1602:
1598:
1566:
1562:
1561:
1559:
1537:
1533:
1523:
1507:
1505:
1502:
1501:
1457:
1453:
1429:
1425:
1418:
1414:
1394:
1391:
1390:
1372:
1366:
1357:
1353:
1349:
1335:
1332:
1327:
1326:
1324:
1319:
1304:
1265:
1261:
1254:
1250:
1246:
1238:
1232:
1227:
1203:
1200:
1199:
1175:
1174:
1151:
1147:
1130:
1126:
1120:
1116:
1109:
1105:
1104:
1098:
1093:
1080:
1076:
1055:
1054:
1029:
1024:
1013:
997:
995:
992:
991:
984:
980:
962:
932:
915:
911:
905:
901:
894:
890:
889:
879:
875:
862:
860:
843:
840:
839:
829:
810:
809:
796:
792:
779:
771:
769:
760:
756:
743:
739:
731:
700:
694:
693:
676:
671:
655:
644:
642:
604:
600:
579:
577:
570:
563:
561:
558:
557:
550:
543:
535:
531:
527:
516:
499:
493:
474:Historically a
472:
464:Earth ellipsoid
451:
441:
439:
429:
426:
424:
415:
412:
410:
344:de La Condamine
319:Jacques Cassini
303:
300:
297:
296:
294:
243:
239:
236:
234:
203:
171:oblate spheroid
167:
161:
149:arc measurement
121:House of Wisdom
113:
103:
75:spherical Earth
17:
12:
11:
5:
8277:
8267:
8266:
8261:
8256:
8242:
8241:
8234:
8233:External links
8231:
8229:
8228:
8164:
8155:
8119:
8110:
8100:Adams, Oscar S
8092:
8083:
8064:(2): 177–190.
8039:
8018:
8009:
7967:(8): 852–861.
7943:
7931:
7919:
7904:
7881:
7839:
7830:
7811:
7781:
7769:
7758:
7745:
7714:
7701:
7694:
7676:
7650:
7646:978-1286804131
7588:
7580:17centurymaths
7571:Isaac Newton:
7564:
7550:. Washington:
7534:
7513:
7500:
7480:
7453:
7451:
7448:
7446:
7445:
7440:
7435:
7430:
7425:
7420:
7417:Paris meridian
7414:
7409:
7404:
7398:
7396:
7393:
7342:
7341:
7326:
7323:
7320:
7315:
7311:
7304:
7301:
7295:
7292:
7290:
7287:
7283:
7279:
7275:
7272:
7269:
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7257:
7250:
7247:
7241:
7236:
7232:
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7222:
7216:
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7211:
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7200:
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7085:
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7070:
7069:
7055:
7054:
7043:
7040:
7037:
7034:
7031:
7028:
7025:
7021:
7017:
7013:
7009:
7006:
7003:
7000:
6997:
6993:
6989:
6985:
6981:
6978:
6975:
6972:
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6957:
6953:
6950:
6947:
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6937:
6933:
6929:
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6919:
6916:
6894:
6893:
6878:
6875:
6872:
6867:
6863:
6856:
6853:
6847:
6844:
6842:
6839:
6835:
6831:
6827:
6824:
6821:
6818:
6813:
6809:
6802:
6799:
6793:
6788:
6784:
6777:
6774:
6768:
6765:
6763:
6760:
6756:
6752:
6748:
6747:
6744:
6741:
6738:
6733:
6729:
6722:
6719:
6713:
6710:
6708:
6705:
6701:
6697:
6693:
6691:
6688:
6685:
6680:
6676:
6669:
6666:
6660:
6657:
6651:
6648:
6642:
6639:
6637:
6634:
6630:
6626:
6622:
6621:
6607:
6606:
6595:
6592:
6589:
6586:
6583:
6580:
6576:
6572:
6568:
6564:
6561:
6558:
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6552:
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6544:
6540:
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6533:
6530:
6527:
6524:
6520:
6516:
6512:
6508:
6505:
6502:
6499:
6496:
6492:
6488:
6484:
6480:
6477:
6474:
6471:
6428:
6427:
6411:
6406:
6402:
6395:
6392:
6387:
6384:
6364:
6357:
6356:
6345:
6338:
6333:
6329:
6325:
6322:
6317:
6314:
6311:
6306:
6302:
6298:
6295:
6289:
6284:
6280:
6276:
6271:
6268:
6265:
6261:
6233:
6230:
6215:
6214:
6201:
6197:
6192:
6188:
6185:
6182:
6179:
6176:
6173:
6170:
6165:
6161:
6140:
6133:
6122:
6121:
6108:
6104:
6100:
6097:
6092:
6088:
6069:
6068:
6046:
6042:
6038:
6035:
6032:
6030:
6028:
6025:
6017:
6014:
6006:
6003:
6000:
5998:
5993:
5988:
5984:
5983:
5960:, see section
5957:
5951:
5950:
5939:
5933:
5930:
5926:
5920:
5915:
5909:
5906:
5903:
5900:
5897:
5894:
5889:
5886:
5883:
5880:
5877:
5874:
5871:
5868:
5862:
5855:
5850:
5847:
5844:
5840:
5834:
5830:
5827:
5824:
5821:
5818:
5815:
5809:
5804:
5800:
5794:
5790:
5787:
5784:
5781:
5778:
5775:
5769:
5763:
5758:
5740:eccentricities
5726:
5723:
5719:
5716:
5705:
5704:
5693:
5690:
5686:
5683:
5679:
5676:
5673:
5670:
5667:
5664:
5661:
5658:
5655:
5652:
5649:
5643:
5638:
5604:
5603:
5592:
5587:
5582:
5579:
5574:
5570:
5567:
5561:
5556:
5541:Earth quadrant
5537:quarter-circle
5513:
5512:
5508:
5507:
5500:
5489:
5467:
5464:
5452:adding a topic
5447:
5436:
5429:
5410:
5409:
5408:
5406:
5399:
5393:
5390:
5389:
5388:
5370:
5367:
5364:
5361:
5358:
5355:
5352:
5349:
5345:
5342:
5339:
5336:
5333:
5305:
5294:
5281:
5272:
5227:
5226:
5203:
5199:
5196:
5193:
5190:
5184:
5181:
5177:
5173:
5170:
5167:
5164:
5161:
5158:
5155:
5151:
5148:
5145:
5142:
5139:
5136:
5132:
5129:
5126:
5123:
5120:
5117:
5113:
5109:
5106:
5101:
5098:
5095:
5091:
5086:
5082:
5078:
5074:
5070:
5067:
5065:
5063:
5052:
5048:
5045:
5042:
5039:
5035:
5031:
5028:
5025:
5022:
5019:
5016:
5012:
5009:
5006:
5003:
5000:
4997:
4993:
4990:
4987:
4984:
4981:
4978:
4974:
4970:
4967:
4962:
4959:
4956:
4952:
4947:
4943:
4939:
4935:
4931:
4928:
4926:
4924:
4921:
4918:
4915:
4912:
4911:
4887:
4876:
4860:
4857:
4831:
4827:
4823:
4820:
4817:
4814:
4811:
4808:
4805:
4802:
4799:
4794:
4791:
4788:
4785:
4782:
4779:
4746:
4745:
4734:
4728:
4725:
4721:
4717:
4714:
4711:
4708:
4705:
4702:
4699:
4696:
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4388:
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4280:
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4230:
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4119:
4116:
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4097:
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3985:
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3946:
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3900:
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3891:
3884:
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3859:
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3851:
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3817:
3814:
3812:
3808:
3804:
3800:
3798:
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3523:
3519:
3516:
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3486:
3482:
3477:
3471:
3467:
3464:
3461:
3455:
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3417:
3414:
3394:
3393:
3382:
3378:
3375:
3372:
3367:
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3359:
3354:
3350:
3346:
3341:
3337:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3309:
3306:
3303:
3299:
3294:
3291:
3288:
3285:
3282:
3279:
3273:
3270:
3237:
3191:
3190:
3175:
3172:
3169:
3164:
3160:
3153:
3150:
3144:
3141:
3139:
3135:
3131:
3127:
3124:
3121:
3118:
3113:
3109:
3102:
3099:
3093:
3088:
3084:
3077:
3074:
3068:
3065:
3063:
3059:
3055:
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3044:
3041:
3036:
3032:
3025:
3022:
3016:
3013:
3010:
3008:
3004:
3000:
2996:
2994:
2991:
2988:
2983:
2979:
2972:
2969:
2963:
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2945:
2942:
2939:
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2933:
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2925:
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2896:
2890:
2885:
2881:
2874:
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2865:
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2854:
2850:
2846:
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2827:
2826:
2815:
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2806:
2803:
2800:
2797:
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2791:
2786:
2782:
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2750:
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2695:
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2658:
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2573:
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2508:
2504:
2497:
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2467:
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2408:
2405:
2403:
2399:
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2376:
2372:
2365:
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2326:
2322:
2315:
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2303:
2301:
2297:
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2285:
2282:
2279:
2274:
2270:
2263:
2260:
2254:
2249:
2245:
2238:
2235:
2229:
2224:
2220:
2213:
2210:
2204:
2199:
2195:
2188:
2185:
2179:
2176:
2173:
2171:
2167:
2163:
2159:
2158:
2155:
2152:
2149:
2144:
2140:
2133:
2130:
2124:
2119:
2115:
2108:
2105:
2099:
2094:
2090:
2083:
2080:
2074:
2069:
2065:
2058:
2055:
2049:
2046:
2043:
2040:
2038:
2034:
2030:
2026:
2025:
2011:
2010:
1999:
1994:
1990:
1987:
1984:
1981:
1978:
1975:
1970:
1966:
1962:
1959:
1956:
1953:
1950:
1945:
1941:
1937:
1934:
1931:
1928:
1925:
1920:
1916:
1912:
1909:
1906:
1903:
1900:
1895:
1891:
1887:
1884:
1879:
1875:
1870:
1864:
1859:
1855:
1849:
1846:
1843:
1840:
1837:
1813:
1806:
1798:Leonhard Euler
1792:
1789:
1784:
1783:
1768:
1764:
1760:
1757:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1730:
1728:
1724:
1720:
1717:
1714:
1711:
1708:
1705:
1697:
1693:
1689:
1683:
1679:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1651:
1647:
1644:
1641:
1639:
1637:
1633:
1626:
1621:
1613:
1610:
1605:
1601:
1597:
1594:
1589:
1586:
1583:
1580:
1577:
1574:
1569:
1565:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1536:
1532:
1529:
1526:
1524:
1522:
1519:
1516:
1513:
1510:
1509:
1487:
1486:
1475:
1471:
1468:
1465:
1460:
1456:
1452:
1449:
1446:
1443:
1438:
1432:
1428:
1424:
1421:
1417:
1413:
1410:
1407:
1404:
1401:
1398:
1365:
1362:
1301:
1300:
1289:
1285:
1282:
1276:
1273:
1268:
1264:
1257:
1253:
1249:
1245:
1242:
1235:
1230:
1226:
1222:
1219:
1216:
1213:
1210:
1207:
1189:
1188:
1173:
1169:
1166:
1158:
1155:
1150:
1145:
1141:
1138:
1133:
1129:
1123:
1119:
1115:
1112:
1108:
1101:
1096:
1092:
1088:
1083:
1079:
1075:
1072:
1069:
1066:
1063:
1060:
1058:
1056:
1053:
1050:
1046:
1043:
1040:
1037:
1032:
1027:
1023:
1019:
1016:
1014:
1012:
1009:
1006:
1003:
1000:
999:
959:
958:
947:
939:
936:
930:
926:
923:
918:
914:
908:
904:
900:
897:
893:
887:
882:
878:
874:
871:
868:
865:
859:
856:
853:
850:
847:
828:
825:
824:
823:
808:
799:
795:
791:
788:
785:
782:
777:
774:
768:
763:
759:
754:
746:
742:
738:
735:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
701:
699:
696:
695:
692:
685:
682:
679:
675:
670:
664:
661:
658:
653:
650:
647:
641:
638:
634:
630:
627:
624:
621:
618:
615:
612:
607:
603:
598:
592:
588:
585:
582:
576:
573:
571:
569:
566:
565:
492:
489:
471:
468:
450:
447:
445: toises.
407:Paris Meridian
380:Anders Celsius
223:pendulum clock
202:
199:
173:to describe a
163:Main article:
160:
157:
102:
99:
83:astro-geodetic
15:
9:
6:
4:
3:
2:
8276:
8265:
8262:
8260:
8257:
8255:
8252:
8251:
8249:
8240:
8237:
8236:
8225:
8220:
8213:
8209:
8205:
8201:
8197:
8193:
8188:
8183:
8179:
8175:
8168:
8159:
8151:
8147:
8143:
8139:
8135:
8134:
8129:
8123:
8114:
8107:
8106:
8101:
8096:
8087:
8079:
8075:
8071:
8067:
8063:
8059:
8058:
8053:
8049:
8043:
8036:
8032:
8028:
8022:
8013:
8006:
8000:
7996:
7992:
7988:
7984:
7980:
7975:
7970:
7966:
7962:
7961:Astron. Nachr
7958:
7957:Bessel, F. W.
7952:
7950:
7948:
7940:
7935:
7929:
7923:
7916:
7915:
7908:
7902:
7898:
7894:
7892:
7885:
7877:
7873:
7869:
7865:
7861:
7858:(in German).
7857:
7853:
7849:
7848:Bessel, F. W.
7843:
7834:
7827:
7825:
7818:
7816:
7808:
7803:
7800:(in French).
7799:
7795:
7791:
7785:
7778:
7773:
7767:
7762:
7755:
7749:
7742:
7738:
7733:
7729:
7725:
7718:
7711:
7705:
7697:
7691:
7687:
7680:
7671:
7667:
7663:
7662:
7654:
7647:
7643:
7639:
7635:
7634:9781440088650
7631:
7627:
7623:
7617:
7613:
7609:
7608:
7603:
7597:
7595:
7593:
7585:
7581:
7577:
7575:
7568:
7554:. p. 307
7553:
7549:
7545:
7538:
7530:
7526:
7525:
7517:
7503:
7497:
7493:
7492:
7484:
7476:
7471:
7470:
7464:
7458:
7454:
7444:
7441:
7439:
7436:
7434:
7431:
7429:
7426:
7424:
7421:
7418:
7415:
7413:
7410:
7408:
7405:
7403:
7400:
7399:
7392:
7358:
7346:
7324:
7321:
7318:
7313:
7309:
7302:
7299:
7293:
7291:
7285:
7281:
7277:
7270:
7267:
7264:
7259:
7255:
7248:
7245:
7239:
7234:
7230:
7223:
7220:
7214:
7212:
7206:
7202:
7198:
7190:
7187:
7184:
7179:
7175:
7168:
7165:
7159:
7157:
7151:
7147:
7143:
7137:
7134:
7131:
7126:
7122:
7115:
7112:
7106:
7103:
7097:
7094:
7088:
7086:
7080:
7076:
7072:
7060:
7059:
7058:
7041:
7038:
7035:
7032:
7029:
7026:
7023:
7019:
7015:
7011:
7007:
7004:
7001:
6998:
6995:
6991:
6987:
6983:
6979:
6976:
6973:
6970:
6967:
6963:
6959:
6955:
6951:
6948:
6945:
6942:
6939:
6935:
6931:
6927:
6923:
6920:
6917:
6914:
6907:
6906:
6905:
6876:
6873:
6870:
6865:
6861:
6854:
6851:
6845:
6843:
6837:
6833:
6829:
6822:
6819:
6816:
6811:
6807:
6800:
6797:
6791:
6786:
6782:
6775:
6772:
6766:
6764:
6758:
6754:
6750:
6742:
6739:
6736:
6731:
6727:
6720:
6717:
6711:
6709:
6703:
6699:
6695:
6689:
6686:
6683:
6678:
6674:
6667:
6664:
6658:
6655:
6649:
6646:
6640:
6638:
6632:
6628:
6624:
6612:
6611:
6610:
6593:
6590:
6587:
6584:
6581:
6578:
6574:
6570:
6566:
6562:
6559:
6556:
6553:
6550:
6546:
6542:
6538:
6534:
6531:
6528:
6525:
6522:
6518:
6514:
6510:
6506:
6503:
6500:
6497:
6494:
6490:
6486:
6482:
6478:
6475:
6472:
6469:
6462:
6461:
6460:
6457:
6453:
6449:
6442:
6438:
6433:
6404:
6400:
6393:
6390:
6385:
6382:
6375:
6374:
6373:
6370:
6363:
6343:
6331:
6327:
6320:
6315:
6312:
6304:
6300:
6293:
6287:
6282:
6278:
6274:
6269:
6266:
6263:
6259:
6251:
6250:
6249:
6247:
6229:
6199:
6195:
6190:
6183:
6180:
6177:
6171:
6168:
6163:
6159:
6151:
6150:
6149:
6147:
6139:
6132:
6127:
6106:
6102:
6098:
6095:
6090:
6086:
6078:
6077:
6076:
6074:
6044:
6040:
6036:
6033:
6031:
6023:
6015:
6012:
6004:
6001:
5999:
5986:
5974:
5973:
5972:
5969:
5967:
5963:
5956:
5937:
5931:
5928:
5924:
5918:
5913:
5907:
5904:
5898:
5895:
5887:
5884:
5878:
5875:
5872:
5869:
5860:
5848:
5845:
5842:
5838:
5832:
5825:
5822:
5819:
5813:
5807:
5802:
5798:
5792:
5785:
5782:
5779:
5773:
5767:
5756:
5748:
5747:
5746:
5743:
5741:
5724:
5721:
5717:
5714:
5691:
5684:
5681:
5677:
5671:
5668:
5665:
5659:
5653:
5650:
5647:
5636:
5628:
5627:
5626:
5624:
5619:
5617:
5613:
5612:nautical mile
5609:
5590:
5585:
5580:
5577:
5572:
5568:
5565:
5554:
5546:
5545:
5544:
5542:
5538:
5534:
5525:
5520:
5505:
5501:
5498:
5490:
5484:
5480:
5476:
5472:
5468:
5465:
5458:
5457:main category
5454:
5453:
5448:
5445:
5441:
5437:
5434:
5431:
5430:
5424:
5418:
5416:
5411:You can help
5407:
5398:
5397:
5374: metres.
5365:
5362:
5359:
5356:
5353:
5350:
5347:
5343:
5337:
5334:
5324:
5323:
5322:
5315:
5304:
5300:
5293:
5289:
5280:
5271:
5265:
5261:
5241:
5233:
5208: metres,
5201:
5197:
5194:
5191:
5188:
5182:
5179:
5175:
5171:
5168:
5165:
5162:
5159:
5156:
5153:
5149:
5146:
5143:
5140:
5137:
5134:
5130:
5127:
5124:
5121:
5118:
5115:
5111:
5107:
5104:
5096:
5089:
5084:
5080:
5076:
5072:
5068:
5066:
5050:
5046:
5043:
5040:
5037:
5033:
5029:
5026:
5023:
5020:
5017:
5014:
5010:
5007:
5004:
5001:
4998:
4995:
4991:
4988:
4985:
4982:
4979:
4976:
4972:
4968:
4965:
4957:
4950:
4945:
4941:
4937:
4933:
4929:
4927:
4919:
4913:
4902:
4901:
4900:
4898:
4893:
4886:
4882:
4875:
4871:
4866:
4856:
4829:
4825:
4821:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4797:
4792:
4789:
4786:
4783:
4780:
4777:
4764:
4760:
4753:
4751:
4732:
4726:
4723:
4719:
4712:
4709:
4706:
4703:
4694:
4691:
4688:
4685:
4677:
4669:
4666:
4660:
4655:
4652:
4648:
4640:
4639:
4638:
4635:
4625:
4619:
4597:
4594:
4591:
4588:
4584:
4577:
4574:
4568:
4565:
4562:
4559:
4556:
4549:
4546:
4540:
4537:
4529:
4526:
4520:
4517:
4514:
4511:
4508:
4505:
4502:
4495:
4492:
4486:
4483:
4480:
4477:
4461:
4458:
4455:
4451:
4447:
4442:
4438:
4430:
4429:
4428:
4404:
4400:
4397:
4394:
4384:
4377:
4372:
4368:
4357:
4353:
4350:
4347:
4337:
4331:
4327:
4320:
4315:
4310:
4307:
4303:
4295:
4294:
4293:
4290:
4263:
4248:
4244:
4240:
4237:
4234:
4231:
4228:
4225:
4222:
4219:
4216:
4211:
4208:
4205:
4202:
4199:
4196:
4190:
4187:
4179:
4176:
4173:
4170:
4167:
4164:
4159:
4155:
4151:
4148:
4145:
4142:
4139:
4134:
4130:
4126:
4123:
4120:
4117:
4114:
4109:
4105:
4101:
4098:
4095:
4092:
4089:
4084:
4080:
4076:
4073:
4068:
4064:
4049:
4045:
4042:
4039:
4033:
4027:
4021:
4010:
4009:
4008:
4006:
3983:
3980:
3977:
3972:
3968:
3961:
3958:
3952:
3949:
3947:
3940:
3936:
3929:
3926:
3923:
3918:
3914:
3907:
3904:
3898:
3893:
3889:
3882:
3879:
3873:
3870:
3868:
3861:
3857:
3849:
3846:
3843:
3838:
3834:
3827:
3824:
3818:
3815:
3813:
3806:
3802:
3796:
3793:
3790:
3785:
3781:
3774:
3771:
3765:
3762:
3756:
3753:
3747:
3744:
3742:
3735:
3731:
3723:
3717:
3713:
3709:
3706:
3703:
3698:
3694:
3687:
3684:
3678:
3673:
3669:
3662:
3659:
3653:
3650:
3647:
3645:
3638:
3634:
3622:
3621:
3620:
3603:
3599:
3595:
3592:
3589:
3586:
3583:
3580:
3575:
3571:
3567:
3564:
3561:
3558:
3555:
3550:
3546:
3542:
3539:
3536:
3533:
3530:
3525:
3521:
3517:
3514:
3511:
3508:
3505:
3500:
3496:
3492:
3489:
3484:
3480:
3475:
3469:
3465:
3462:
3459:
3453:
3447:
3441:
3434:
3433:
3432:
3430:
3423:
3413:
3411:
3407:
3403:
3380:
3373:
3370:
3365:
3361:
3357:
3352:
3348:
3344:
3339:
3335:
3331:
3328:
3325:
3322:
3316:
3313:
3307:
3304:
3301:
3297:
3292:
3286:
3283:
3280:
3271:
3268:
3258:
3257:
3256:
3245:
3241:
3236:
3229:
3225:
3173:
3170:
3167:
3162:
3158:
3151:
3148:
3142:
3140:
3133:
3129:
3122:
3119:
3116:
3111:
3107:
3100:
3097:
3091:
3086:
3082:
3075:
3072:
3066:
3064:
3057:
3053:
3045:
3042:
3039:
3034:
3030:
3023:
3020:
3014:
3011:
3009:
3002:
2998:
2992:
2989:
2986:
2981:
2977:
2970:
2967:
2961:
2958:
2952:
2949:
2943:
2940:
2938:
2931:
2927:
2919:
2916:
2913:
2908:
2904:
2897:
2894:
2888:
2883:
2879:
2872:
2869:
2863:
2860:
2857:
2855:
2848:
2844:
2832:
2831:
2830:
2813:
2808:
2804:
2801:
2798:
2795:
2792:
2789:
2784:
2780:
2776:
2773:
2770:
2767:
2764:
2759:
2755:
2751:
2748:
2745:
2742:
2739:
2734:
2730:
2726:
2723:
2720:
2717:
2714:
2709:
2705:
2701:
2698:
2693:
2689:
2684:
2678:
2674:
2671:
2668:
2662:
2656:
2650:
2643:
2642:
2641:
2639:
2635:
2616:
2607:
2599:
2596:
2593:
2585:
2582:
2576:
2571:
2567:
2559:
2558:
2557:
2552:
2538:
2517:
2514:
2511:
2506:
2502:
2495:
2492:
2486:
2484:
2477:
2473:
2465:
2462:
2459:
2454:
2450:
2443:
2440:
2434:
2429:
2425:
2418:
2415:
2409:
2406:
2404:
2397:
2393:
2385:
2382:
2379:
2374:
2370:
2363:
2360:
2354:
2349:
2345:
2338:
2335:
2329:
2324:
2320:
2313:
2310:
2304:
2302:
2295:
2291:
2283:
2280:
2277:
2272:
2268:
2261:
2258:
2252:
2247:
2243:
2236:
2233:
2227:
2222:
2218:
2211:
2208:
2202:
2197:
2193:
2186:
2183:
2177:
2174:
2172:
2165:
2161:
2153:
2150:
2147:
2142:
2138:
2131:
2128:
2122:
2117:
2113:
2106:
2103:
2097:
2092:
2088:
2081:
2078:
2072:
2067:
2063:
2056:
2053:
2047:
2044:
2041:
2039:
2032:
2028:
2016:
2015:
2014:
1997:
1992:
1988:
1985:
1982:
1979:
1976:
1973:
1968:
1964:
1960:
1957:
1954:
1951:
1948:
1943:
1939:
1935:
1932:
1929:
1926:
1923:
1918:
1914:
1910:
1907:
1904:
1901:
1898:
1893:
1889:
1885:
1882:
1877:
1873:
1868:
1862:
1857:
1853:
1847:
1841:
1835:
1828:
1827:
1826:
1823:
1818:
1805:
1803:
1799:
1788:
1766:
1758:
1755:
1751:
1748:
1745:
1739:
1736:
1733:
1731:
1722:
1715:
1712:
1709:
1703:
1695:
1691:
1687:
1681:
1677:
1671:
1665:
1662:
1659:
1653:
1649:
1645:
1642:
1640:
1631:
1624:
1619:
1611:
1608:
1603:
1599:
1595:
1592:
1587:
1584:
1581:
1578:
1575:
1572:
1567:
1563:
1556:
1550:
1547:
1544:
1538:
1534:
1530:
1527:
1525:
1517:
1511:
1500:
1499:
1498:
1496:
1492:
1473:
1466:
1463:
1458:
1454:
1450:
1447:
1436:
1430:
1426:
1422:
1419:
1415:
1411:
1408:
1402:
1396:
1389:
1388:
1387:
1385:
1381:
1377:
1371:
1361:
1346:
1339:
1330:
1322:
1316:
1312:
1308:
1287:
1283:
1280:
1274:
1271:
1266:
1262:
1255:
1251:
1247:
1243:
1240:
1233:
1228:
1224:
1220:
1217:
1211:
1205:
1198:
1197:
1196:
1194:
1171:
1167:
1164:
1156:
1153:
1148:
1143:
1139:
1136:
1131:
1127:
1121:
1117:
1113:
1110:
1106:
1099:
1094:
1090:
1081:
1077:
1073:
1070:
1064:
1061:
1059:
1051:
1048:
1041:
1035:
1030:
1025:
1021:
1017:
1015:
1007:
1001:
990:
989:
988:
977:
973:
969:
965:
945:
937:
934:
928:
924:
921:
916:
912:
906:
902:
898:
895:
891:
880:
876:
872:
869:
863:
857:
851:
845:
838:
837:
836:
834:
806:
797:
789:
786:
783:
775:
772:
766:
761:
757:
752:
744:
740:
736:
733:
728:
725:
719:
716:
713:
707:
704:
702:
697:
690:
683:
680:
677:
673:
668:
662:
659:
656:
651:
648:
645:
639:
636:
632:
625:
622:
619:
613:
610:
605:
601:
596:
590:
586:
583:
580:
574:
572:
567:
556:
555:
554:
549:
541:
524:
522:
514:
513:
508:
504:
498:
488:
486:
482:
477:
476:nautical mile
467:
465:
461:
457:
446:
437:
422:
408:
404:
400:
396:
392:
387:
385:
381:
377:
373:
369:
365:
361:
357:
353:
349:
345:
341:
337:
333:
329:
325:
320:
316:
312:
292:
288:
284:
283:
278:
273:
271:
267:
263:
259:
254:
250:
232:
231:French Guiana
228:
224:
220:
216:
212:
208:
198:
196:
192:
188:
184:
180:
176:
172:
166:
156:
154:
150:
146:
142:
138:
134:
130:
126:
122:
118:
112:
108:
98:
96:
92:
88:
84:
80:
76:
71:
69:
65:
61:
57:
52:
50:
46:
42:
38:
34:
30:
26:
22:
8180:(1): 43–55.
8177:
8173:
8167:
8158:
8137:
8132:
8122:
8113:
8104:
8095:
8086:
8061:
8055:
8042:
8034:
8021:
8012:
8004:
7964:
7960:
7934:
7922:
7912:
7907:
7890:
7884:
7859:
7855:
7842:
7833:
7823:
7801:
7797:
7784:
7772:
7761:
7748:
7737:Maxima files
7723:
7717:
7704:
7685:
7679:
7660:
7653:
7606:
7573:
7567:
7556:. Retrieved
7547:
7537:
7523:
7516:
7505:. Retrieved
7490:
7483:
7468:
7463:Russo, Lucio
7457:
7371:replaced by
7363:replaced by
7351:in terms of
7343:
7056:
6900:in terms of
6895:
6608:
6458:
6451:
6447:
6440:
6436:
6429:
6368:
6361:
6358:
6248:, iterating
6240:, determine
6235:
6216:
6137:
6130:
6123:
6070:
5970:
5954:
5952:
5744:
5706:
5620:
5616:hebdomometre
5605:
5540:
5532:
5530:
5479:edit summary
5470:
5450:
5420:
5412:
5321:is given by
5313:
5302:
5298:
5291:
5287:
5278:
5269:
5266:
5259:
5239:
5231:
5228:
5057: metres
4894:
4884:
4880:
4873:
4869:
4862:
4762:
4758:
4754:
4747:
4636:
4617:
4614:
4426:
4291:
4287:
4002:
3618:
3419:
3395:
3246:
3239:
3234:
3227:
3223:
3192:
2828:
2631:
2548:
2536:
2012:
1821:
1815:
1794:
1785:
1488:
1373:
1347:
1337:
1328:
1320:
1314:
1310:
1306:
1302:
1190:
975:
971:
967:
963:
960:
830:
540:eccentricity
525:
511:
510:
500:
485:day skippers
473:
452:
449:19th century
388:
376:Abbe Outhier
323:
317:and his son
280:
277:Isaac Newton
274:
268:than at the
253:acceleration
204:
194:
190:
181:in place of
178:
170:
168:
140:
133:Eratosthenes
114:
72:
67:
53:
47:, or to its
29:meridian arc
28:
18:
7622:Archive.org
6226:.146 m
5966:James Ivory
5610:and of the
4632:(−3)!! = −1
491:Calculation
460:Clarke 1866
456:Bessel 1841
340:Louis Godin
311:Jean Picard
215:Jean Richer
129:Alexandrian
8248:Categories
7804:: 258–293.
7638:Nabu Press
7558:2009-01-28
7507:2021-05-02
7450:References
6071:The polar
5517:See also:
5422:(May 2021)
4628:(−1)!! = 1
1382:handbook (
827:Definition
548:flattening
495:See also:
372:Le Monnier
360:Maupertuis
356:to Lapland
352:Jorge Juan
291:flattening
145:Posidonius
131:scientist
25:navigation
8212:119310141
8187:1109.4448
8150:312469983
8078:251572677
8048:Ivory, J.
7999:118760590
7974:0908.1824
7790:Euler, L.
7574:Principia
7322:⋯
7319:−
7268:⋯
7240:−
7188:⋯
7185:−
7135:⋯
7107:−
7039:⋯
7033:μ
7027:
7005:μ
6999:
6977:μ
6971:
6949:μ
6943:
6921:μ
6915:β
6874:⋯
6820:⋯
6792:−
6740:⋯
6687:⋯
6659:−
6594:⋯
6588:μ
6582:
6560:μ
6554:
6532:μ
6526:
6504:μ
6498:
6476:μ
6470:φ
6391:π
6383:μ
6328:φ
6313:−
6301:φ
6288:−
6279:φ
6260:φ
6126:perimeter
6013:π
5876:−
5854:∞
5839:∑
5814:π
5774:π
5578:π
5497:talk page
5449:Consider
5417:in German
5366:φ
5360:
5351:−
5332:Δ
5198:β
5192:
5180:−
5172:×
5166:−
5163:β
5157:
5147:−
5144:β
5138:
5128:−
5125:β
5119:
5105:−
5097:∘
5090:β
5047:φ
5041:
5024:φ
5018:
5008:−
5005:φ
4999:
4986:φ
4980:
4966:−
4958:∘
4951:φ
4920:φ
4819:φ
4813:
4793:φ
4787:
4689:−
4667:−
4518:−
4484:−
4467:∞
4452:∑
4238:φ
4232:
4212:φ
4206:
4191:−
4180:⋯
4177:−
4174:φ
4168:
4149:φ
4143:
4127:−
4124:φ
4118:
4099:φ
4093:
4077:−
4074:φ
4028:φ
3981:⋯
3953:−
3927:⋯
3874:−
3847:⋯
3819:−
3794:⋯
3748:−
3707:⋯
3596:⋯
3590:β
3584:
3565:β
3559:
3540:β
3534:
3515:β
3509:
3490:β
3448:φ
3429:geodesics
3374:⋯
3371:−
3345:−
3326:−
3171:⋯
3168:−
3120:⋯
3117:−
3092:−
3043:⋯
3015:−
2990:⋯
2944:−
2917:⋯
2805:⋯
2799:φ
2793:
2774:φ
2768:
2749:φ
2743:
2724:φ
2718:
2699:φ
2657:φ
2632:In 1837,
2515:⋯
2463:⋯
2460:−
2435:−
2410:−
2383:⋯
2281:⋯
2278:−
2253:−
2228:−
2203:−
2178:−
2151:⋯
1989:⋯
1983:φ
1977:
1958:φ
1952:
1933:φ
1927:
1908:φ
1902:
1883:φ
1842:φ
1804:squared.
1746:β
1710:φ
1692:φ
1660:φ
1625:φ
1612:
1596:−
1588:φ
1585:
1579:φ
1576:
1557:−
1545:φ
1518:φ
1448:φ
1442:Π
1423:−
1403:φ
1284:β
1275:β
1272:
1234:β
1225:∫
1212:φ
1168:φ
1149:−
1140:φ
1137:
1114:−
1100:φ
1091:∫
1074:−
1052:φ
1042:φ
1031:φ
1022:∫
1008:φ
925:φ
922:
899:−
873:−
852:φ
737:−
717:−
681:−
649:−
623:−
584:−
293:equal to
282:Principia
275:In 1687,
211:spherical
195:ellipsoid
187:ellipsoid
153:Al-Ma'mun
37:longitude
8130:(1811).
8102:(1921).
8050:(1798).
7850:(1837).
7792:(1755).
7604:(1880).
7584:pdf file
7465:(2004).
7395:See also
7286:′
7207:′
7152:′
7081:′
7020:′
6992:′
6964:′
6936:′
6838:′
6759:′
6704:′
6633:′
6575:′
6547:′
6519:′
6491:′
6050: m.
5725:′
5685:′
5473:provide
4391:if
4344:if
3408:and the
3244:vanish.
3193:Because
1817:Delambre
1759:′
1252:′
364:Clairaut
287:spheroid
258:latitude
191:Spheroid
183:spheroid
45:meridian
8254:Geodesy
8224:Addenda
8192:Bibcode
7979:Bibcode
7864:Bibcode
7807:Figures
7616:2484948
7607:Geodesy
7491:Geodesy
7407:Geodesy
6430:is the
6045:965.729
5495:to the
5477:in the
5419:.
5250:
5236:
5112:346.170
5081:132.952
4973:038.509
4942:132.952
4761:)(1 + 2
4622:is the
3404:by the
3220:
3208:
2638:Helmert
1342:
1325:
1309:= (1 −
438:bar as
432: m
397:to the
395:Dunkirk
336:Bouguer
324:prolate
307:
295:
270:Equator
242:⁄
227:Cayenne
219:gravity
125:Baghdad
43:of the
41:segment
31:is the
21:geodesy
8210:
8148:
8076:
8037:, 1–13
7997:
7777:Maxima
7692:
7670:906501
7668:
7644:
7632:
7614:
7498:
7057:where
6609:where
6372:where
6021:
6008:
5707:where
5229:where
4992:16.833
4757:(1 − 2
4427:where
2496:131072
2013:where
1303:where
979:(with
421:toises
419:
354:) and
175:sphere
137:stadia
117:caliph
91:WGS 84
49:length
8208:S2CID
8182:arXiv
8136:[
8074:S2CID
7995:S2CID
7969:arXiv
7739:and
7477:-277.
7359:with
5543:, is
5440:DeepL
5150:0.001
5131:1.122
5030:0.000
5011:0.022
4897:WGS84
3619:with
2829:with
2364:16384
2132:16384
2129:11025
1313:)tan
440:0.513
436:metre
401:(the
368:Camus
249:Paris
93:(see
64:geoid
33:curve
8146:OCLC
7690:ISBN
7666:OCLC
7642:ISBN
7630:ISBN
7624:and
7612:OCLC
7496:ISBN
7383:and
7303:1536
6852:1097
6148:is:
6136:= 2π
6124:The
5471:must
5469:You
5433:View
5297:) −
5276:and
5234:° =
4879:) −
4630:and
4615:and
4398:>
3201:and
2444:4096
2419:3072
2361:2205
2339:1024
2262:4096
2259:2205
2237:1024
1380:NIST
1336:1 −
1323:′ =
1318:and
1305:tan
831:The
442:0762
193:and
109:and
27:, a
23:and
8200:doi
8142:180
8066:doi
7987:doi
7965:331
7897:doi
7872:doi
7728:doi
7475:273
7300:539
7024:sin
6996:sin
6968:sin
6940:sin
6855:512
6718:151
6579:sin
6551:sin
6523:sin
6495:sin
6224:449
6221:367
6172:0.5
6041:001
5442:or
5357:cos
5354:560
5348:133
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5264:).
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5189:sin
5169:0.5
5154:sin
5135:sin
5116:sin
5077:111
5038:sin
5015:sin
4996:sin
4977:sin
4938:111
4810:cos
4784:sin
4229:cos
4203:sin
4165:sin
4140:sin
4115:sin
4090:sin
4007:as
3962:512
3581:sin
3556:sin
3531:sin
3506:sin
3251:or
3152:512
3149:315
2790:sin
2765:sin
2740:sin
2715:sin
2493:315
2441:105
2336:105
2314:256
2234:525
2107:256
2104:175
1974:sin
1949:sin
1924:sin
1899:sin
1609:sin
1582:cos
1573:sin
1497:),
1386:),
1263:sin
1128:sin
987:is
913:sin
430:000
427:000
416:762
413:130
304:230
289:of
225:to
123:in
119:'s
97:).
19:In
8250::
8206:.
8198:.
8190:.
8178:87
8176:.
8144:.
8072:.
8060:.
8054:.
8035:59
8033:,
7993:.
7985:.
7977:.
7963:.
7946:^
7870:.
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7814:^
7726:,
7591:^
7546:.
7529:20
7391:.
7249:96
7246:37
7224:16
7169:96
7166:29
7116:32
6801:32
6798:55
6776:16
6773:21
6721:96
6668:32
6665:27
6367:=
6228:.
6037:10
5968:.
5742:.
5625:,
5618:.
5247:1°
5176:10
5085:55
5034:03
4969:16
4946:55
4855:.
4634:.
4620:!!
3908:64
3883:16
3828:48
3775:16
3688:64
3431:,
3412:.
3226:+
3101:64
3098:15
3076:16
3073:15
3024:48
3021:35
2971:16
2898:64
2640:,
2416:35
2311:15
2212:32
2209:15
2082:64
2079:45
1825:,
1352:,
1345:.
1195:,
976:dφ
974:)
966:=
964:dm
542:,
534:,
530:,
523:.
466:.
425:10
378:,
374:,
370:,
366:,
362:,
350:,
346:,
342:,
338:,
272:.
260:,
229:,
155:.
51:.
8226:.
8214:.
8202::
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8080:.
8068::
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7874::
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7730::
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7672:.
7640:(
7628:(
7618:.
7561:.
7531:.
7510:.
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7385:τ
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7377:ε
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7369:β
7365:s
7361:m
7353:β
7349:m
7325:.
7314:4
7310:n
7294:=
7282:8
7278:B
7271:,
7265:+
7260:4
7256:n
7235:2
7231:n
7221:5
7215:=
7203:4
7199:B
7191:,
7180:3
7176:n
7160:=
7148:6
7144:B
7138:,
7132:+
7127:3
7123:n
7113:9
7104:n
7098:2
7095:1
7089:=
7077:2
7073:B
7042:,
7036:+
7030:8
7016:8
7012:B
7008:+
7002:6
6988:6
6984:B
6980:+
6974:4
6960:4
6956:B
6952:+
6946:2
6932:2
6928:B
6924:+
6918:=
6902:β
6898:m
6877:.
6871:+
6866:4
6862:n
6846:=
6834:8
6830:H
6823:,
6817:+
6812:4
6808:n
6787:2
6783:n
6767:=
6755:4
6751:H
6743:,
6737:+
6732:3
6728:n
6712:=
6700:6
6696:H
6690:,
6684:+
6679:3
6675:n
6656:n
6650:2
6647:3
6641:=
6629:2
6625:H
6591:+
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6571:8
6567:H
6563:+
6557:6
6543:6
6539:H
6535:+
6529:4
6515:4
6511:H
6507:+
6501:2
6487:2
6483:H
6479:+
6473:=
6454:)
6452:φ
6450:(
6448:M
6443:)
6441:φ
6439:(
6437:m
6410:p
6405:m
6401:m
6394:2
6386:=
6369:μ
6365:0
6362:φ
6344:,
6337:)
6332:i
6324:(
6321:M
6316:m
6310:)
6305:i
6297:(
6294:m
6283:i
6275:=
6270:1
6267:+
6264:i
6242:φ
6238:m
6219:6
6200:0
6196:c
6191:/
6187:)
6184:b
6181:+
6178:a
6175:(
6169:=
6164:r
6160:M
6141:r
6138:M
6134:p
6131:C
6107:p
6103:m
6099:4
6096:=
6091:p
6087:C
6034:=
6024:a
6016:2
6002:=
5992:p
5987:m
5958:0
5955:c
5938:,
5932:j
5929:2
5925:n
5919:2
5914:)
5908:!
5905:!
5902:)
5899:j
5896:2
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5888:!
5885:!
5882:)
5879:3
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5870:2
5867:(
5861:(
5849:0
5846:=
5843:j
5833:4
5829:)
5826:b
5823:+
5820:a
5817:(
5808:=
5803:0
5799:c
5793:4
5789:)
5786:b
5783:+
5780:a
5777:(
5768:=
5762:p
5757:m
5722:e
5718:,
5715:e
5692:.
5689:)
5682:e
5678:i
5675:(
5672:E
5669:b
5666:=
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5660:e
5657:(
5654:E
5651:a
5648:=
5642:p
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5591:.
5586:)
5581:2
5573:(
5569:m
5566:=
5560:p
5555:m
5506:.
5499:.
5369:)
5363:2
5341:(
5338:=
5335:m
5319:φ
5314:m
5312:Δ
5308:)
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5301:(
5299:m
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5292:φ
5290:(
5288:m
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5260:β
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5232:φ
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4930:=
4923:)
4917:(
4914:m
4890:)
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4885:φ
4883:(
4881:m
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4872:(
4870:m
4853:β
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4826:n
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4807:n
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4763:k
4759:k
4733:.
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4716:)
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4701:(
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4695:k
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4683:(
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4661:=
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4649:H
4618:k
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4575:!
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4509:+
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4503:2
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4493:!
4490:)
4487:3
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4475:(
4462:0
4459:=
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4448:=
4443:k
4439:c
4405:,
4401:0
4395:k
4385:,
4378:k
4373:k
4369:c
4358:,
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4351:=
4348:k
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4332:0
4328:c
4321:{
4316:=
4311:k
4308:2
4304:B
4264:.
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4241:+
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4171:8
4160:8
4156:B
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4110:4
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4096:2
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4069:0
4065:B
4057:(
4050:2
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4043:+
4040:a
4034:=
4031:)
4025:(
4022:m
3984:.
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3973:4
3969:n
3959:5
3950:=
3941:8
3937:B
3930:,
3924:+
3919:4
3915:n
3905:1
3899:+
3894:2
3890:n
3880:1
3871:=
3862:4
3858:B
3850:,
3844:+
3839:3
3835:n
3825:1
3816:=
3807:6
3803:B
3797:,
3791:+
3786:3
3782:n
3772:1
3766:+
3763:n
3757:2
3754:1
3745:=
3736:2
3732:B
3724:,
3718:0
3714:H
3710:=
3704:+
3699:4
3695:n
3685:1
3679:+
3674:2
3670:n
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3660:1
3654:+
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3648:=
3639:0
3635:B
3604:,
3600:)
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3576:8
3572:B
3568:+
3562:6
3551:6
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3493:+
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3476:(
3470:2
3466:b
3463:+
3460:a
3454:=
3451:)
3445:(
3442:m
3425:β
3398:n
3381:,
3377:)
3366:4
3362:n
3358:+
3353:3
3349:n
3340:2
3336:n
3332:+
3329:n
3323:1
3320:(
3317:a
3314:=
3308:n
3305:+
3302:1
3298:a
3293:=
3290:)
3287:b
3284:+
3281:a
3278:(
3272:2
3269:1
3253:b
3249:a
3240:k
3238:2
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3230:)
3228:b
3224:a
3222:(
3217:2
3214:/
3211:1
3203:b
3199:a
3195:n
3174:.
3163:4
3159:n
3143:=
3134:8
3130:H
3123:,
3112:4
3108:n
3087:2
3083:n
3067:=
3058:4
3054:H
3046:,
3040:+
3035:3
3031:n
3012:=
3003:6
2999:H
2993:,
2987:+
2982:3
2978:n
2968:3
2962:+
2959:n
2953:2
2950:3
2941:=
2932:2
2928:H
2920:,
2914:+
2909:4
2905:n
2895:1
2889:+
2884:2
2880:n
2873:4
2870:1
2864:+
2861:1
2858:=
2849:0
2845:H
2814:,
2809:)
2802:+
2796:8
2785:8
2781:H
2777:+
2771:6
2760:6
2756:H
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2735:4
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2710:2
2706:H
2702:+
2694:0
2690:H
2685:(
2679:2
2675:b
2672:+
2669:a
2663:=
2660:)
2654:(
2651:m
2617:.
2608:2
2604:)
2600:n
2597:+
2594:1
2591:(
2586:n
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2577:=
2572:2
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2518:.
2512:+
2507:8
2503:e
2487:=
2478:8
2474:D
2466:,
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2426:e
2407:=
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2394:D
2386:,
2380:+
2375:8
2371:e
2355:+
2350:6
2346:e
2330:+
2325:4
2321:e
2305:=
2296:4
2292:D
2284:,
2273:8
2269:e
2248:6
2244:e
2223:4
2219:e
2198:2
2194:e
2187:8
2184:3
2175:=
2166:2
2162:D
2154:,
2148:+
2143:8
2139:e
2123:+
2118:6
2114:e
2098:+
2093:4
2089:e
2073:+
2068:2
2064:e
2057:4
2054:3
2048:+
2045:1
2042:=
2033:0
2029:D
1998:,
1993:)
1986:+
1980:8
1969:8
1965:D
1961:+
1955:6
1944:6
1940:D
1936:+
1930:4
1919:4
1915:D
1911:+
1905:2
1894:2
1890:D
1886:+
1878:0
1874:D
1869:(
1863:a
1858:2
1854:b
1848:=
1845:)
1839:(
1836:m
1822:e
1812:)
1810:e
1767:.
1763:)
1756:e
1752:i
1749:,
1743:(
1740:E
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1734:=
1723:)
1719:)
1716:e
1713:,
1707:(
1704:E
1696:2
1688:d
1682:2
1678:d
1672:+
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1666:e
1663:,
1657:(
1654:E
1650:(
1646:a
1643:=
1632:)
1620:2
1604:2
1600:e
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1551:e
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1539:E
1535:(
1531:a
1528:=
1521:)
1515:(
1512:m
1474:.
1470:)
1467:e
1464:,
1459:2
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1445:(
1437:)
1431:2
1427:e
1420:1
1416:(
1412:a
1409:=
1406:)
1400:(
1397:m
1358:μ
1354:β
1350:φ
1338:e
1333:/
1329:e
1321:e
1315:φ
1311:f
1307:β
1288:,
1281:d
1267:2
1256:2
1248:e
1244:+
1241:1
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1221:b
1218:=
1215:)
1209:(
1206:m
1172:.
1165:d
1157:2
1154:3
1144:)
1132:2
1122:2
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1111:1
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1095:0
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1065:a
1062:=
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1039:(
1036:M
1026:0
1018:=
1011:)
1005:(
1002:m
985:φ
981:φ
972:φ
970:(
968:M
946:,
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935:3
929:)
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892:(
886:)
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870:1
867:(
864:a
858:=
855:)
849:(
846:M
807:.
798:2
794:)
790:n
787:+
784:1
781:(
776:n
773:4
767:=
762:2
758:e
753:,
745:2
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734:1
729:a
726:=
723:)
720:f
714:1
711:(
708:a
705:=
698:b
691:,
684:f
678:2
674:f
669:=
663:b
660:+
657:a
652:b
646:a
640:=
637:n
633:,
629:)
626:f
620:2
617:(
614:f
611:=
606:2
602:e
597:,
591:a
587:b
581:a
575:=
568:f
551:n
544:e
536:f
532:b
528:a
517:φ
411:5
358:(
334:(
301:/
298:1
244:2
240:1
237:+
235:2
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