2543:
2030:
8230:
5235:
5413:
5535:
2538:{\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}}}
4009:
3199:
4916:
6902:
7350:
1792:
4289:
3636:
2846:
5230:{\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}}}
832:
1197:
6626:
7074:
4004:{\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}}}
4024:
1514:
3194:{\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}}}
2019:
2835:
4299:
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.
5959:
489:
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
3625:
6077:
570:
1004:
6897:{\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}}}
7345:{\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}}}
332:
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
4284:{\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}}}
1787:{\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}}}
4433:
6615:
7063:
416:). 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
967:
4621:
1309:
6365:
1842:
3402:
2657:
464:
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,
2035:
5762:
1806:
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,
1797:
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.
5457:, 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.
4856:
3448:
1495:
827:{\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}}}
5988:
5397:
1192:{\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}}}
6436:
5612:
4029:
266:
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
4309:
1359:
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
5713:
4754:
7079:
6631:
5993:
4921:
3641:
2851:
1519:
1009:
575:
4878:. 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
498:: "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".
2638:
4303:
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
6223:
150:, 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
6476:
6921:
5443:
852:
6130:
4444:
5467:
5747:
393:). The resulting measurements at equatorial and polar latitudes confirmed that the Earth was best modelled by an oblate spheroid, supporting Newton. However, by 1743,
77:
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
2014:{\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)\,,}
7906:, 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
1212:
6265:
2830:{\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)\,,}
3272:
5954:{\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}\,,}
7358:
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
7559:
United States
National Museum Bulletin 240: Contributions from the Museum of History and Technology reprinted in Bulletin of the Smithsonian Institution
7672:
Comparisons of the standards of length of
England, France, Belgium, Prussia, Russia, India, Australia, made at the Ordnance survey office, Southampton
3620:{\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),}
5477:
Do not translate text that appears unreliable or low-quality. If possible, verify the text with references provided in the foreign-language article.
4781:
1403:
7746:
Section 5.6. This reference includes the derivation of curvature formulae from first principles and a proof of
Meusnier's theorem. (Supplements:
92:
in the region the survey was to be conducted, leading to a proliferation of reference ellipsoids around the world. The latest determinations use
6072:{\displaystyle {\begin{aligned}m_{\mathrm {p} }&=0.9983242984312529\ {\frac {\pi }{2}}\ a\\&=10\,001\,965.729{\mbox{ m.}}\end{aligned}}}
8067:
7386:, the arc length on the auxiliary sphere. The requisite series extended to sixth order are given by Charles Karney, Eqs. (17) & (21), with
4014:
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
292:
7863:"Bestimmung der Axen des elliptischen Rotationssphäroids, welches den vorhandenen Messungen von Meridianbögen der Erde am meisten entspricht"
5633:
5338:
6389:
5560:
8041:
1501:
1386:
4428:{\displaystyle B_{2k}={\begin{cases}c_{0}\,,&{\text{if }}k=0\,,\\{\dfrac {c_{k}}{k}}\,,&{\text{if }}k>0\,,\end{cases}}}
8038:
A General
Formula for Calculating Meridian Arc Length and its Application to Coordinate Conversion in the Gauss-Krüger Projection
3416:
413:
342:
7510:
7949:
4654:
7704:
5462:
7656:
7433:
5485:
366:
100:
to determine reference ellipsoids, especially the geocentric ellipsoids now used for global coordinate systems such as
7670:
2573:
7644:
5642:
5498:
Content in this edit is translated from the existing German
Knowledge article at ]; see its history for attribution.
3420:
8274:
17:
5750:
1812:
8269:
3412:
3411:. Even though this results in more slowly converging series, such series are used in the specification for the
531:
378:
7648:
7562:
4875:
1827:
474:
401:
354:
4334:
126:
Early estimations of Earth's size are recorded from Greece in the 4th century BC, and from scholars at the
6165:
2561:
8058:
6610:{\displaystyle \varphi =\mu +H'_{2}\sin 2\mu +H'_{4}\sin 4\mu +H'_{6}\sin 6\mu +H'_{8}\sin 8\mu +\cdots }
5976:
7776:
7058:{\displaystyle \beta =\mu +B'_{2}\sin 2\mu +B'_{4}\sin 4\mu +B'_{6}\sin 6\mu +B'_{8}\sin 8\mu +\cdots ,}
962:{\displaystyle M(\varphi )={\frac {a(1-e^{2})}{\left(1-e^{2}\sin ^{2}\varphi \right)^{\frac {3}{2}}}},}
550:
491:
338:
272:
8114:
7938:
7453:
7367:
4616:{\displaystyle c_{k}=\sum _{j=0}^{\infty }{\frac {(2j-3)!!\,(2j+2k-3)!!}{(2j)!!\,(2j+2k)!!}}n^{k+2j}}
3439:
516:. On an ellipsoid of revolution, for short meridian arcs, their length can be approximated using the
382:
325:
8229:
7937:
J. W. Hager, J.F. Behensky, and B.W. Drew, 1989. Defense
Mapping Agency Technical Report TM 8358.2.
1203:
7485:
6083:
5493:
4760:
2648:
7817:
7554:
6092:
520:
and the circular arc formulation. For longer arcs, the length follows from the subtraction of two
8249:
7438:
6156:
5618:
370:
362:
7939:
The universal grids: Universal
Transverse Mercator (UTM) and Universal Polar Stereographic (UPS)
175:
8062:
7804:
8138:
7900:
7805:"Élémens de la trigonométrie sphéroïdique tirés de la méthode des plus grands et plus petits"
7612:
7355:
5514:
4907:
470:
394:
8152:
7865:[Estimation of the axes of the ellipsoid through measurements of the meridian arc].
7583:
7500:
7477:
1304:{\displaystyle m(\varphi )=b\int _{0}^{\beta }{\sqrt {1+e'^{2}\sin ^{2}\beta }}\,d\beta \,,}
513:
224:, by the 17th century, evidence was accumulating that it was not a perfect sphere. In 1672,
8202:
7989:
7874:
7833:
7636:
7589:, translated into English by Andrew Motte. A searchable modern translation is available at
7539:
6360:{\displaystyle \varphi _{i+1}=\varphi _{i}-{\frac {m(\varphi _{i})-m}{M(\varphi _{i})}}\,,}
4858:
appearing in the above formula and results in poorer convergence of the series in terms of
337:
spheroid (with an equatorial radius less than the polar radius). To resolve the issue, the
121:
88:
required a single arc. Accurate survey work beginning in the 19th century required several
55:
7807:[Elements of spheroidal trigonometry taken from the method of maxima and minima].
6139:
of a meridian ellipse can also be rewritten in the form of a rectifying circle perimeter,
5721:
564:. Only two of these parameters are independent and there are many relations between them:
8:
7478:
7448:
7443:
7422:
6442:
3432:
3397:{\displaystyle {\tfrac {1}{2}}(a+b)={\frac {a}{1+n}}=a(1-n+n^{2}-n^{3}+n^{4}-\cdots )\,,}
2560:
Series with considerably faster convergence can be obtained by expanding in terms of the
217:
70:
66:
8206:
7993:
7878:
7594:
386:
8218:
8192:
8144:
Exercices de Calcul Intégral sur Divers Ordres de Transcendantes et sur les Quadratures
8084:
8005:
7979:
7970:(2010). "The calculation of longitude and latitude from geodesic measurements (1825)".
7412:
6256:
232:
was not constant over the Earth (as it would be if the Earth were a sphere); he took a
117:
93:
7616:
843:
517:
8222:
8156:
8088:
8009:
7700:
7676:
7675:. London: G.E. Eyre and W. Spottiswoode for H.M. Stationery Office. pp. 281–87.
7652:
7640:
7622:
7506:
5507:
5489:
4015:
409:
276:
97:
549:, but in theoretical work it is useful to define extra parameters, particularly the
8264:
8210:
8076:
7997:
7967:
7907:
7882:
7858:
7738:
4634:
2644:
466:
434:
as specified by the standard toise bar in Paris. Defining this distance as exactly
358:
89:
43:
8142:
7533:
6470:
Alternatively, Helmert's series for the meridian distance can be reverted to give
530:. This is an important problem in the theory of map projections, particularly the
69:. One or more measurements of meridian arcs can be used to infer the shape of the
374:
329:
221:
159:
158:
about 150 years later, and slightly better results were calculated in 827 by the
131:
85:
8234:
7924:
507:
8110:
7800:
7632:
7427:
5547:
5529:
5425:
1808:
417:
390:
346:
320:. This was disputed by some, but not all, French scientists. A meridian arc of
233:
147:
51:
8250:
Online computation of meridian arcs on different geodetic reference ellipsoids
8214:
8080:
7763:
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors, 2010,
3431:
In 1825, Bessel derived an expansion of the meridian distance in terms of the
8258:
8160:
7886:
5622:
5534:
4648:
The coefficients in Helmert's series can similarly be expressed generally by
486:
241:
208:
are used interchangeably in this article, with oblate implied if not stated.
8037:
5982:
The numerical expression for the quarter meridian on the WGS84 ellipsoid is
4851:{\displaystyle {\frac {2n\sin 2\varphi }{\sqrt {1+2n\cos 2\varphi +n^{2}}}}}
1490:{\displaystyle m(\varphi )=a\left(1-e^{2}\right)\,\Pi (\varphi ,e^{2},e)\,.}
1380:
8001:
7626:
5626:
994:
in radians). Therefore, the meridian distance from the equator to latitude
287:
263:
163:
143:
7680:
7911:
7742:
7473:
6247:
In some problems, we need to be able to solve the inverse problem: given
4874:
The trigonometric series given above can be conveniently evaluated using
495:
350:
321:
225:
5756:
The quarter meridian is also given by the following generalized series:
4906:
Substituting the values for the semi-major axis and eccentricity of the
3243:
is constant under this interchange, half the terms in the expansions of
196:, although the qualifying words "of revolution" are usually dropped. An
5496:
to the source of your translation. A model attribution edit summary is
558:
301:
200:
that is not an ellipsoid of revolution is called a triaxial ellipsoid.
155:
139:
59:
35:
5392:{\displaystyle \Delta m=(111\,133-560\cos 2\varphi ){\mbox{ metres.}}}
6136:
197:
146:
about 240 BC. He estimated that the meridian has a length of 252,000
47:
7902:
Die mathematischen und physikalischen Theorieen der höheren Geodäsie
7862:
7720:
7366:
and its inverse given above play a key role in the solution of the
6431:{\displaystyle \mu ={\frac {\pi }{2}}{\frac {m}{m_{\mathrm {p} }}}}
5607:{\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,.}
297:
268:
193:
8197:
8119:. US Coast and Geodetic Survey Special Publication No. 67. p. 127.
7984:
7751:
7747:
7417:
6445:. Note that it there is no need to differentiate the series for
6242:
405:
280:
237:
229:
135:
31:
7555:"Paper 44: Development of gravity pendulums in the 19th century"
5454:
138:
in the 9th century. The first realistic value was calculated by
7835:
Méthodes Analytiques pour la Détermination d'un Arc du Méridien
5328:
between the two parallels at ±0.5° from the circle at latitude
2647:
obtained one such series, which was put into a simpler form by
1505:
1394:
185:
127:
101:
972:
The arc length of an infinitesimal element of the meridian is
7837:; précédées d'un mémoire sur le même sujet par A. M. Legendre
3426:
1202:
The distance formula is simpler when written in terms of the
446:
431:
259:
154:, which has not been preserved. A similar method was used by
74:
8116:
Latitude Developments Connected With Geodesy and Cartography
7590:
7531:
7928:. Royal Prussian Geodetic Institute, New Series 52, page 12
4421:
1390:
7552:
6370:
until convergence. A suitable starting guess is given by
5466:
to this template: there are already 1,879 articles in the
4637:, extended to negative values via the recursion relation:
46:
between two points on the Earth's surface having the same
27:
Distance along a portion of a meridian, for use in geodesy
7809:
Mémoires de l'Académie Royale des Sciences de Berlin 1753
7659:), the first chapter covers the history of early surveys.
6456:, since the formula for the meridian radius of curvature
5278:
On the ellipsoid the exact distance between parallels at
2551:
2548:
Richard Rapp gives a detailed derivation of this result.
526:, the distance from the equator to a point at a latitude
188:"squashed at the poles". Modern literature uses the term
65:
The purpose of measuring meridian arcs is to determine a
7787:
7764:
5321:. For WGS84 an approximate expression for the distance
7699:. Hamble: The Royal Yachting Association. p. 76.
7308:
7254:
7229:
7174:
7121:
7103:
6860:
6806:
6781:
6726:
6673:
6655:
6059:
5632:
The quarter meridian can be expressed in terms of the
5383:
5217:
5066:
3967:
3913:
3888:
3833:
3780:
3762:
3693:
3668:
3277:
3157:
3106:
3081:
3029:
2976:
2958:
2903:
2878:
2501:
2449:
2424:
2369:
2344:
2319:
2267:
2242:
2217:
2192:
2137:
2112:
2087:
2062:
1385:
The above integral is related to a special case of an
8183:
Karney, C. F. F. (2013). "Algorithms for geodesics".
7492:
7077:
6924:
6629:
6479:
6392:
6268:
6168:
6095:
5991:
5765:
5724:
5645:
5563:
5341:
4919:
4784:
4657:
4447:
4376:
4312:
4027:
3639:
3451:
3275:
2849:
2660:
2576:
2033:
1845:
1818:
1517:
1406:
1215:
1007:
855:
573:
8063:"A new series for the rectification of the ellipsis"
5450:
4749:{\displaystyle H_{2k}=(-1)^{k}(1-2k)(1+2k)B_{2k}\,.}
473:. A comprehensive list of ellipsoids is given under
7532:Poynting, John Henry; Joseph John Thompson (1907).
5446:
a machine-translated version of the German article.
512:On a sphere, the meridian arc length is simply the
105:
7790:, 2009, A computer algebra system, version 5.20.1.
7344:
7057:
6896:
6609:
6430:
6359:
6217:
6124:
6071:
5953:
5741:
5707:
5606:
5391:
5229:
4850:
4748:
4615:
4427:
4283:
4003:
3619:
3396:
3193:
2829:
2632:
2537:
2013:
1786:
1489:
1303:
1191:
961:
826:
7925:Konforme Abbildung des Erdellipsoids in der Ebene
4269:
4067:
3216:are interchanged, and because the initial factor
2567:instead of the eccentricity. They are related by
1629:
1374:
8256:
2633:{\displaystyle e^{2}={\frac {4n}{(1+n)^{2}}}\,.}
1502:incomplete elliptic integrals of the second kind
404:had remeasured and extended the French arc from
5708:{\displaystyle m_{\mathrm {p} }=aE(e)=bE(ie').}
5542:The distance from the equator to the pole, the
3266:as the initial factor by writing, for example,
8068:Transactions of the Royal Society of Edinburgh
7950:A guide to coordinate systems in Great Britain
7647:). In addition the book has been reprinted by
7505:. De Gruyter Textbook. De Gruyter. p. 5.
6243:The inverse meridian problem for the ellipsoid
5492:accompanying your translation by providing an
5437:Click for important translation instructions.
5424:expand this section with text translated from
1387:incomplete elliptic integral of the third kind
7669:Clarke, Alexander Ross; James, Henry (1866).
7553:Victor F., Lenzen; Robert P. Multauf (1964).
5634:complete elliptic integral of the second kind
397:had completely supplanted Newton's approach.
84:The earliest determinations of the size of a
8151:] (in French). Paris: Courcier. p.
7538:. London: Charles Griffin & Co. p.
4778:originates from the additional expansion of
7962:
7960:
7958:
7839:, De L'Imprimerie de Crapelet, Paris, 72–73
7828:
7826:
7668:
7498:
1830:in 1799 derived a widely used expansion on
490:from the latitude scale of charts. As the
7466:
5975:above.) This result was first obtained by
4903:while maintaining high relative accuracy.
3427:Series in terms of the parametric latitude
445:led to the construction of a new standard
211:
8196:
8104:
8042:Geospatial Information Authority of Japan
7983:
7694:
7546:
6353:
6054:
6050:
5947:
5600:
5357:
5198:
5163:
5144:
5125:
5121:
5098:
5094:
5090:
5047:
5043:
5024:
5005:
4986:
4982:
4959:
4955:
4951:
4742:
4563:
4509:
4414:
4394:
4367:
4347:
3733:
3390:
2823:
2626:
2007:
1776:
1483:
1451:
1297:
1290:
1181:
1174:
1058:
816:
762:
700:
642:
606:
111:
8137:
7955:
7823:
7607:
7605:
7603:
7587:, Book III, Proposition XIX, Problem III
5533:
5268:expressed in degrees (and similarly for
4869:
3407:and expanding the result as a series in
3258:The series can be expressed with either
296:as a proof that the Earth was an oblate
258:minutes per day compared to its rate at
7765:NIST Handbook of Mathematical Functions
7732:
7726:
7378:, the distance along the geodesic, and
6086:is simply four times quarter meridian:
5972:
3417:National Geospatial-Intelligence Agency
14:
8257:
8182:
8014:English translation of Astron. Nachr.
7966:
7857:
7611:
518:Earth's meridional radius of curvature
8057:
7799:
7777:Mathematica guide: Elliptic Integrals
7600:
7576:
7472:
5538:A quarter meridian or Earth quadrant.
4294:
537:The main ellipsoidal parameters are,
7943:
6218:{\displaystyle M_{r}=0.5(a+b)/c_{0}}
5625:, and used in the definition of the
5406:
2552:Expansions in the third flattening (
1801:
480:
414:meridian arc of Delambre and Méchain
176:Earth ellipsoid § Determination
169:
7952:, Ordnance Survey of Great Britain.
5402:
1500:It may also be written in terms of
24:
7434:French Geodesic Mission to Lapland
6420:
6002:
5864:
5772:
5652:
5570:
5342:
4477:
1452:
81:intended to fit the entire world.
25:
8286:
8243:
7848:Rapp, R, (1991), §3.6, pp. 36–40.
7430:(West Europe-Africa Meridian-arc)
216:Although it had been known since
162:method, attributed to the Caliph
50:. The term may refer either to a
8228:
7685:Appendix on Figure of the Earth.
5411:
4763:and proved by Kazushige Kawase.
3421:Ordnance Survey of Great Britain
1819:Expansions in the eccentricity (
1389:. In the notation of the online
324:was extended to a longer arc by
275:being about 0.5% greater at the
96:measurements and the methods of
8176:
8167:
8131:
8122:
8095:
8051:
8030:
8021:
7931:
7916:
7893:
7851:
7842:
7793:
7781:
7770:
7757:
7697:RYA day skipper handbook - sail
7284:
6907:Similarly, Bessel's series for
6836:
4759:This result was conjectured by
4200:
3943:
3438:in connection with his work on
3136:
766:
646:
610:
459:
180:Early literature uses the term
8149:Exercises in Integral Calculus
7713:
7688:
7662:
7525:
7499:Torge, W.; Müller, J. (2012).
6347:
6334:
6320:
6307:
6197:
6185:
5912:
5903:
5892:
5877:
5839:
5827:
5799:
5787:
5699:
5685:
5673:
5667:
5617:It was part of the historical
5502:You may also add the template
5379:
5351:
5110:
5104:
4971:
4965:
4933:
4927:
4726:
4711:
4708:
4693:
4684:
4674:
4582:
4564:
4554:
4545:
4534:
4510:
4500:
4485:
4041:
4035:
3461:
3455:
3413:transverse Mercator projection
3387:
3330:
3300:
3288:
2670:
2664:
2614:
2601:
1855:
1849:
1773:
1753:
1729:
1717:
1679:
1667:
1564:
1552:
1531:
1525:
1480:
1455:
1416:
1410:
1375:Relation to elliptic integrals
1367:, and the rectifying latitude
1225:
1219:
1097:
1078:
1055:
1049:
1021:
1015:
896:
877:
865:
859:
846:can be shown to be equal to:
804:
791:
733:
721:
639:
627:
532:transverse Mercator projection
501:
228:found the first evidence that
13:
1:
7767:(Cambridge University Press).
7631:. Freely available online at
7563:Smithsonian Institution Press
7535:A Textbook of Physics, 4th Ed
7460:
5504:{{Translated|de|Erdquadrant}}
837:
402:Jean Baptiste Joseph Delambre
7484:. Berlin: Springer. p.
6125:{\displaystyle C_{p}=4m_{p}}
5530:Ellipse § Circumference
1811:derived an expansion in the
844:meridian radius of curvature
508:Latitude § Meridian arc
7:
7832:Delambre, J. B. J. (1799):
7621:. Oxford: Clarendon Press.
7405:
5474:will aid in categorization.
400:By the end of the century,
152:On the measure of the Earth
73:that best approximates the
10:
8291:
5527:
5449:Machine translation, like
1378:
505:
492:Royal Yachting Association
339:French Academy of Sciences
273:gravitational acceleration
173:
115:
8215:10.1007/s00190-012-0578-z
8081:10.1017/s0080456800030817
7867:Astronomische Nachrichten
7721:Geometric Geodesy, Part I
7454:Geodesics on an ellipsoid
5749:are the first and second
5426:the corresponding article
1381:Ellipse § Arc length
326:Giovanni Domenico Cassini
7887:10.1002/asna.18370142301
7735:The Mercator Projections
7695:Hopkinson, Sara (2012).
7480:The Forgotten Revolution
6915:can be reverted to give
6255:. This may be solved by
8173:Helmert (1880), Chap. 5
7904:, Einleitung und 1 Teil
7899:Helmert, F. R. (1880):
7733:Osborne, Peter (2013),
7593:. Search the following
7439:French Geodesic Mission
6228:It can be evaluated as
6157:rectifying Earth radius
5619:definition of the metre
5513:For more guidance, see
4862:compared to the one in
1504:(See the NIST handbook
494:says in its manual for
244:and found that it lost
212:17th and 18th centuries
190:ellipsoid of revolution
8275:History of measurement
8002:10.1002/asna.201011352
7752:Latex code and figures
7613:Clarke, Alexander Ross
7346:
7059:
6898:
6611:
6432:
6361:
6219:
6126:
6073:
5955:
5868:
5743:
5709:
5608:
5539:
5393:
5231:
4852:
4750:
4617:
4481:
4429:
4285:
4005:
3621:
3398:
3195:
2831:
2634:
2539:
2015:
1788:
1491:
1305:
1193:
963:
828:
262:. This indicated the
112:History of measurement
8270:Meridians (geography)
8101:Helmert (1880), §1.10
8027:Helmert (1880), §1.11
8018:, 241–254 (1825), §5.
7356:Adrien-Marie Legendre
7347:
7060:
6899:
6612:
6467:can be used instead.
6433:
6362:
6220:
6127:
6084:Earth's circumference
6074:
5956:
5848:
5744:
5710:
5609:
5550:), also known as the
5537:
5515:Knowledge:Translation
5486:copyright attribution
5394:
5232:
4870:Numerical expressions
4853:
4751:
4618:
4461:
4430:
4286:
4006:
3622:
3399:
3196:
2832:
2635:
2540:
2016:
1789:
1492:
1379:Further information:
1306:
1194:
964:
829:
290:had published in the
116:Further information:
106:numerical expressions
8128:Helmert (1880), §5.6
7912:10.5281/zenodo.32050
7743:10.5281/zenodo.35392
7723:, §3.5.1, pp. 28–32.
7398:playing the role of
7390:playing the role of
7075:
6922:
6627:
6477:
6390:
6266:
6166:
6093:
5989:
5964:(For the formula of
5763:
5742:{\displaystyle e,e'}
5722:
5643:
5561:
5339:
4917:
4782:
4655:
4445:
4310:
4025:
3637:
3449:
3273:
2847:
2658:
2574:
2031:
1843:
1515:
1404:
1371:, are unrestricted.
1213:
1005:
853:
571:
469:, Everest 1830, and
122:History of the metre
79:geocentric ellipsoid
8207:2013JGeod..87...43K
8036:Kawase, K. (2011):
7994:2010AN....331..852K
7922:Krüger, L. (1912):
7879:1837AN.....14..333B
7449:Rectifying latitude
7444:Struve Geodetic Arc
7423:Reference ellipsoid
7299:
7220:
7165:
7094:
7033:
7005:
6977:
6949:
6851:
6772:
6717:
6646:
6588:
6560:
6532:
6504:
6443:rectifying latitude
5973:#Generalized series
3433:parametric latitude
1248:
1204:parametric latitude
1114:
1045:
514:circular arc length
343:expeditions to Peru
220:that the Earth was
218:classical antiquity
71:reference ellipsoid
67:figure of the Earth
8185:Journal of Geodesy
8040:, Bulletin of the
7413:History of geodesy
7342:
7340:
7317:
7287:
7263:
7238:
7208:
7183:
7153:
7130:
7112:
7082:
7055:
7021:
6993:
6965:
6937:
6894:
6892:
6869:
6839:
6815:
6790:
6760:
6735:
6705:
6682:
6664:
6634:
6607:
6576:
6548:
6520:
6492:
6428:
6357:
6215:
6122:
6069:
6067:
6063:
6016:0.9983242984312529
5951:
5739:
5705:
5604:
5546:(analogous to the
5540:
5494:interlanguage link
5389:
5387:
5227:
5225:
5221:
5070:
4876:Clenshaw summation
4848:
4746:
4613:
4425:
4420:
4392:
4295:Generalized series
4281:
4279:
4001:
3999:
3976:
3922:
3897:
3842:
3789:
3771:
3702:
3677:
3617:
3394:
3286:
3208:changes sign when
3191:
3189:
3166:
3115:
3090:
3038:
2985:
2967:
2912:
2887:
2827:
2630:
2535:
2533:
2510:
2458:
2433:
2378:
2353:
2328:
2276:
2251:
2226:
2201:
2146:
2121:
2096:
2071:
2011:
1813:third eccentricity
1784:
1782:
1487:
1301:
1234:
1189:
1187:
1100:
1031:
959:
824:
822:
523:meridian distances
420:was calculated as
395:Clairaut's theorem
277:geographical poles
118:History of geodesy
7719:Rapp, R, (1991):
7512:978-3-11-025000-8
7316:
7262:
7237:
7182:
7129:
7111:
6868:
6814:
6789:
6734:
6681:
6663:
6426:
6407:
6351:
6155:. Therefore, the
6062:
6033:
6029:
6020:
5922:
5846:
5806:
5594:
5526:
5525:
5438:
5434:
5386:
5220:
5069:
4846:
4845:
4766:The extra factor
4761:Friedrich Helmert
4592:
4403:
4391:
4356:
4265:
4264:
4063:
4016:geodetic latitude
3975:
3921:
3896:
3841:
3788:
3770:
3701:
3676:
3483:
3322:
3285:
3165:
3114:
3089:
3037:
2984:
2966:
2911:
2886:
2692:
2624:
2509:
2457:
2432:
2377:
2352:
2327:
2275:
2250:
2225:
2200:
2145:
2120:
2095:
2070:
1876:
1802:Series expansions
1712:
1639:
1638:
1288:
1170:
954:
951:
814:
760:
698:
677:
604:
481:The nautical mile
410:Mediterranean Sea
341:(1735) undertook
170:Ellipsoidal Earth
98:satellite geodesy
16:(Redirected from
8282:
8238:
8233:
8232:
8226:
8200:
8180:
8174:
8171:
8165:
8164:
8135:
8129:
8126:
8120:
8108:
8102:
8099:
8093:
8092:
8055:
8049:
8034:
8028:
8025:
8019:
8013:
7987:
7964:
7953:
7947:
7941:
7935:
7929:
7920:
7914:
7897:
7891:
7890:
7873:(333): 333–346.
7855:
7849:
7846:
7840:
7830:
7821:
7816:
7797:
7791:
7785:
7779:
7774:
7768:
7761:
7755:
7745:
7730:
7724:
7717:
7711:
7710:
7692:
7686:
7684:
7666:
7660:
7630:
7609:
7598:
7580:
7574:
7573:
7571:
7570:
7550:
7544:
7543:
7529:
7523:
7522:
7520:
7519:
7496:
7490:
7489:
7483:
7470:
7401:
7397:
7393:
7389:
7385:
7381:
7377:
7373:
7368:geodesic problem
7365:
7361:
7351:
7349:
7348:
7343:
7341:
7328:
7327:
7318:
7309:
7295:
7274:
7273:
7264:
7255:
7249:
7248:
7239:
7230:
7216:
7194:
7193:
7184:
7175:
7161:
7141:
7140:
7131:
7122:
7113:
7104:
7090:
7064:
7062:
7061:
7056:
7029:
7001:
6973:
6945:
6914:
6910:
6903:
6901:
6900:
6895:
6893:
6880:
6879:
6870:
6861:
6847:
6826:
6825:
6816:
6807:
6801:
6800:
6791:
6782:
6768:
6746:
6745:
6736:
6727:
6713:
6693:
6692:
6683:
6674:
6665:
6656:
6642:
6616:
6614:
6613:
6608:
6584:
6556:
6528:
6500:
6466:
6455:
6437:
6435:
6434:
6429:
6427:
6425:
6424:
6423:
6410:
6408:
6400:
6382:
6366:
6364:
6363:
6358:
6352:
6350:
6346:
6345:
6329:
6319:
6318:
6302:
6297:
6296:
6284:
6283:
6254:
6250:
6238:
6236:
6233:
6224:
6222:
6221:
6216:
6214:
6213:
6204:
6178:
6177:
6154:
6131:
6129:
6128:
6123:
6121:
6120:
6105:
6104:
6078:
6076:
6075:
6070:
6068:
6064:
6060:
6040:
6031:
6030:
6022:
6018:
6007:
6006:
6005:
5960:
5958:
5957:
5952:
5946:
5945:
5933:
5932:
5927:
5923:
5921:
5901:
5875:
5867:
5862:
5847:
5842:
5822:
5817:
5816:
5807:
5802:
5782:
5777:
5776:
5775:
5748:
5746:
5745:
5740:
5738:
5714:
5712:
5711:
5706:
5698:
5657:
5656:
5655:
5613:
5611:
5610:
5605:
5599:
5595:
5587:
5575:
5574:
5573:
5544:quarter meridian
5505:
5499:
5473:
5472:|topic=
5470:, and specifying
5455:Google Translate
5436:
5432:
5415:
5414:
5407:
5403:Quarter meridian
5398:
5396:
5395:
5390:
5388:
5384:
5331:
5327:
5320:
5295:
5286:
5274:
5267:
5263:
5262:
5260:
5259:
5256:
5253:
5236:
5234:
5233:
5228:
5226:
5222:
5218:
5215:
5211:
5197:
5196:
5114:
5113:
5075:
5071:
5067:
5064:
5060:
4975:
4974:
4910:ellipsoid gives
4902:
4865:
4861:
4857:
4855:
4854:
4849:
4847:
4844:
4843:
4807:
4806:
4786:
4777:
4755:
4753:
4752:
4747:
4741:
4740:
4692:
4691:
4670:
4669:
4644:
4640:
4635:double factorial
4632:
4622:
4620:
4619:
4614:
4612:
4611:
4593:
4591:
4543:
4483:
4480:
4475:
4457:
4456:
4434:
4432:
4431:
4426:
4424:
4423:
4404:
4401:
4393:
4387:
4386:
4377:
4357:
4354:
4346:
4345:
4325:
4324:
4290:
4288:
4287:
4282:
4280:
4273:
4272:
4266:
4263:
4262:
4226:
4225:
4205:
4196:
4174:
4173:
4149:
4148:
4124:
4123:
4099:
4098:
4083:
4082:
4071:
4070:
4064:
4059:
4048:
4010:
4008:
4007:
4002:
4000:
3987:
3986:
3977:
3968:
3955:
3954:
3933:
3932:
3923:
3914:
3908:
3907:
3898:
3889:
3876:
3875:
3853:
3852:
3843:
3834:
3821:
3820:
3800:
3799:
3790:
3781:
3772:
3763:
3750:
3749:
3732:
3731:
3713:
3712:
3703:
3694:
3688:
3687:
3678:
3669:
3653:
3652:
3626:
3624:
3623:
3618:
3613:
3609:
3590:
3589:
3565:
3564:
3540:
3539:
3515:
3514:
3499:
3498:
3484:
3479:
3468:
3437:
3410:
3403:
3401:
3400:
3395:
3380:
3379:
3367:
3366:
3354:
3353:
3323:
3321:
3307:
3287:
3278:
3265:
3261:
3254:
3242:
3232:
3230:
3229:
3226:
3223:
3215:
3211:
3207:
3200:
3198:
3197:
3192:
3190:
3177:
3176:
3167:
3158:
3148:
3147:
3126:
3125:
3116:
3107:
3101:
3100:
3091:
3082:
3072:
3071:
3049:
3048:
3039:
3030:
3017:
3016:
2996:
2995:
2986:
2977:
2968:
2959:
2946:
2945:
2923:
2922:
2913:
2904:
2898:
2897:
2888:
2879:
2863:
2862:
2836:
2834:
2833:
2828:
2822:
2818:
2799:
2798:
2774:
2773:
2749:
2748:
2724:
2723:
2708:
2707:
2693:
2688:
2677:
2645:Friedrich Bessel
2639:
2637:
2636:
2631:
2625:
2623:
2622:
2621:
2599:
2591:
2586:
2585:
2566:
2562:third flattening
2555:
2544:
2542:
2541:
2536:
2534:
2521:
2520:
2511:
2502:
2492:
2491:
2469:
2468:
2459:
2450:
2444:
2443:
2434:
2425:
2412:
2411:
2389:
2388:
2379:
2370:
2364:
2363:
2354:
2345:
2339:
2338:
2329:
2320:
2310:
2309:
2287:
2286:
2277:
2268:
2262:
2261:
2252:
2243:
2237:
2236:
2227:
2218:
2212:
2211:
2202:
2193:
2180:
2179:
2157:
2156:
2147:
2138:
2132:
2131:
2122:
2113:
2107:
2106:
2097:
2088:
2082:
2081:
2072:
2063:
2047:
2046:
2020:
2018:
2017:
2012:
2006:
2002:
1983:
1982:
1958:
1957:
1933:
1932:
1908:
1907:
1892:
1891:
1877:
1872:
1871:
1862:
1835:
1822:
1793:
1791:
1790:
1785:
1783:
1772:
1740:
1736:
1732:
1713:
1711:
1710:
1709:
1696:
1695:
1686:
1649:
1645:
1641:
1640:
1634:
1633:
1627:
1618:
1617:
1602:
1601:
1582:
1581:
1571:
1506:Section 19.6(iv)
1496:
1494:
1493:
1488:
1473:
1472:
1450:
1446:
1445:
1444:
1395:Section 19.2(ii)
1370:
1366:
1362:
1355:
1354:
1352:
1351:
1345:
1342:
1328:
1310:
1308:
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1242:
1198:
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1195:
1190:
1188:
1173:
1172:
1171:
1163:
1157:
1153:
1146:
1145:
1136:
1135:
1113:
1108:
1096:
1095:
1068:
1044:
1039:
997:
993:
989:
968:
966:
965:
960:
955:
953:
952:
944:
942:
938:
931:
930:
921:
920:
899:
895:
894:
872:
833:
831:
830:
825:
823:
815:
813:
812:
811:
789:
781:
776:
775:
761:
759:
758:
743:
699:
697:
683:
678:
676:
665:
654:
620:
619:
605:
600:
589:
563:
557:, and the third
556:
548:
544:
540:
529:
455:
454:
444:
442:
439:
429:
428:
425:
359:Antonio de Ulloa
319:
317:
316:
313:
310:
257:
256:
252:
249:
90:arc measurements
21:
8290:
8289:
8285:
8284:
8283:
8281:
8280:
8279:
8255:
8254:
8246:
8241:
8227:
8181:
8177:
8172:
8168:
8139:Legendre, A. M.
8136:
8132:
8127:
8123:
8109:
8105:
8100:
8096:
8056:
8052:
8035:
8031:
8026:
8022:
7965:
7956:
7948:
7944:
7936:
7932:
7921:
7917:
7898:
7894:
7856:
7852:
7847:
7843:
7831:
7824:
7798:
7794:
7786:
7782:
7775:
7771:
7762:
7758:
7731:
7727:
7718:
7714:
7707:
7706:9781-9051-04949
7693:
7689:
7667:
7663:
7637:Forgotten Books
7610:
7601:
7597:for 'spheroid'.
7581:
7577:
7568:
7566:
7551:
7547:
7530:
7526:
7517:
7515:
7513:
7497:
7493:
7471:
7467:
7463:
7458:
7408:
7399:
7395:
7391:
7387:
7383:
7379:
7375:
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7359:
7339:
7338:
7323:
7319:
7307:
7300:
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7285:
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7265:
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7212:
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7189:
7185:
7173:
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7157:
7151:
7136:
7132:
7120:
7102:
7095:
7086:
7078:
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7025:
6997:
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6891:
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6330:
6314:
6310:
6303:
6301:
6292:
6288:
6273:
6269:
6267:
6264:
6263:
6257:Newton's method
6252:
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6245:
6234:
6231:
6229:
6209:
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6200:
6173:
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6167:
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2581:
2577:
2575:
2572:
2571:
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2553:
2532:
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2512:
2500:
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2407:
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2400:
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2384:
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2334:
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2282:
2278:
2266:
2257:
2253:
2241:
2232:
2228:
2216:
2207:
2203:
2191:
2181:
2175:
2171:
2168:
2167:
2152:
2148:
2136:
2127:
2123:
2111:
2102:
2098:
2086:
2077:
2073:
2061:
2048:
2042:
2038:
2034:
2032:
2029:
2028:
1978:
1974:
1953:
1949:
1928:
1924:
1903:
1899:
1887:
1883:
1882:
1878:
1867:
1863:
1861:
1844:
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1825:
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1804:
1781:
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1765:
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1663:
1659:
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1628:
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1609:
1577:
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1570:
1548:
1544:
1534:
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1516:
1513:
1512:
1468:
1464:
1440:
1436:
1429:
1425:
1405:
1402:
1401:
1383:
1377:
1368:
1364:
1360:
1346:
1343:
1338:
1337:
1335:
1330:
1315:
1276:
1272:
1265:
1261:
1257:
1249:
1243:
1238:
1214:
1211:
1210:
1186:
1185:
1162:
1158:
1141:
1137:
1131:
1127:
1120:
1116:
1115:
1109:
1104:
1091:
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1065:
1040:
1035:
1024:
1008:
1006:
1003:
1002:
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991:
973:
943:
926:
922:
916:
912:
905:
901:
900:
890:
886:
873:
871:
854:
851:
850:
840:
821:
820:
807:
803:
790:
782:
780:
771:
767:
754:
750:
742:
711:
705:
704:
687:
682:
666:
655:
653:
615:
611:
590:
588:
581:
574:
572:
569:
568:
561:
554:
546:
542:
538:
527:
510:
504:
485:Historically a
483:
475:Earth ellipsoid
462:
452:
450:
440:
437:
435:
426:
423:
421:
355:de La Condamine
330:Jacques Cassini
314:
311:
308:
307:
305:
254:
250:
247:
245:
214:
182:oblate spheroid
178:
172:
160:arc measurement
132:House of Wisdom
124:
114:
86:spherical Earth
28:
23:
22:
18:Meridional line
15:
12:
11:
5:
8288:
8278:
8277:
8272:
8267:
8253:
8252:
8245:
8244:External links
8242:
8240:
8239:
8175:
8166:
8130:
8121:
8111:Adams, Oscar S
8103:
8094:
8075:(2): 177–190.
8050:
8029:
8020:
7978:(8): 852–861.
7954:
7942:
7930:
7915:
7892:
7850:
7841:
7822:
7792:
7780:
7769:
7756:
7725:
7712:
7705:
7687:
7661:
7657:978-1286804131
7599:
7591:17centurymaths
7582:Isaac Newton:
7575:
7561:. Washington:
7545:
7524:
7511:
7491:
7464:
7462:
7459:
7457:
7456:
7451:
7446:
7441:
7436:
7431:
7428:Paris meridian
7425:
7420:
7415:
7409:
7407:
7404:
7353:
7352:
7337:
7334:
7331:
7326:
7322:
7315:
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7306:
7303:
7301:
7298:
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7054:
7051:
7048:
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7032:
7028:
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7020:
7017:
7014:
7011:
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7004:
7000:
6996:
6992:
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6986:
6983:
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6976:
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6964:
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6955:
6952:
6948:
6944:
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6930:
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6889:
6886:
6883:
6878:
6874:
6867:
6864:
6858:
6855:
6853:
6850:
6846:
6842:
6838:
6835:
6832:
6829:
6824:
6820:
6813:
6810:
6804:
6799:
6795:
6788:
6785:
6779:
6776:
6774:
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6767:
6763:
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6758:
6755:
6752:
6749:
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6733:
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6724:
6721:
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6712:
6708:
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6699:
6696:
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6662:
6659:
6653:
6650:
6648:
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6641:
6637:
6633:
6632:
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6617:
6606:
6603:
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6594:
6591:
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6538:
6535:
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6527:
6523:
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6510:
6507:
6503:
6499:
6495:
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6488:
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6422:
6417:
6413:
6406:
6403:
6398:
6395:
6375:
6368:
6367:
6356:
6349:
6344:
6340:
6336:
6333:
6328:
6325:
6322:
6317:
6313:
6309:
6306:
6300:
6295:
6291:
6287:
6282:
6279:
6276:
6272:
6244:
6241:
6226:
6225:
6212:
6208:
6203:
6199:
6196:
6193:
6190:
6187:
6184:
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6176:
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6132:
6119:
6115:
6111:
6108:
6103:
6099:
6080:
6079:
6057:
6053:
6049:
6046:
6043:
6041:
6039:
6036:
6028:
6025:
6017:
6014:
6011:
6009:
6004:
5999:
5995:
5994:
5971:, see section
5968:
5962:
5961:
5950:
5944:
5941:
5937:
5931:
5926:
5920:
5917:
5914:
5911:
5908:
5905:
5900:
5897:
5894:
5891:
5888:
5885:
5882:
5879:
5873:
5866:
5861:
5858:
5855:
5851:
5845:
5841:
5838:
5835:
5832:
5829:
5826:
5820:
5815:
5811:
5805:
5801:
5798:
5795:
5792:
5789:
5786:
5780:
5774:
5769:
5751:eccentricities
5737:
5734:
5730:
5727:
5716:
5715:
5704:
5701:
5697:
5694:
5690:
5687:
5684:
5681:
5678:
5675:
5672:
5669:
5666:
5663:
5660:
5654:
5649:
5615:
5614:
5603:
5598:
5593:
5590:
5585:
5581:
5578:
5572:
5567:
5552:Earth quadrant
5548:quarter-circle
5524:
5523:
5519:
5518:
5511:
5500:
5478:
5475:
5463:adding a topic
5458:
5447:
5440:
5421:
5420:
5419:
5417:
5410:
5404:
5401:
5400:
5399:
5381:
5378:
5375:
5372:
5369:
5366:
5363:
5360:
5356:
5353:
5350:
5347:
5344:
5316:
5305:
5292:
5283:
5238:
5237:
5214:
5210:
5207:
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5195:
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5175:
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5166:
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5156:
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5150:
5147:
5143:
5140:
5137:
5134:
5131:
5128:
5124:
5120:
5117:
5112:
5109:
5106:
5102:
5097:
5093:
5089:
5085:
5081:
5078:
5076:
5074:
5063:
5059:
5056:
5053:
5050:
5046:
5042:
5039:
5036:
5033:
5030:
5027:
5023:
5020:
5017:
5014:
5011:
5008:
5004:
5001:
4998:
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4992:
4989:
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4978:
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4963:
4958:
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4950:
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4942:
4939:
4937:
4935:
4932:
4929:
4926:
4923:
4922:
4898:
4887:
4871:
4868:
4842:
4838:
4834:
4831:
4828:
4825:
4822:
4819:
4816:
4813:
4810:
4805:
4802:
4799:
4796:
4793:
4790:
4757:
4756:
4745:
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2300:
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2265:
2260:
2256:
2249:
2246:
2240:
2235:
2231:
2224:
2221:
2215:
2210:
2206:
2199:
2196:
2190:
2187:
2184:
2182:
2178:
2174:
2170:
2169:
2166:
2163:
2160:
2155:
2151:
2144:
2141:
2135:
2130:
2126:
2119:
2116:
2110:
2105:
2101:
2094:
2091:
2085:
2080:
2076:
2069:
2066:
2060:
2057:
2054:
2051:
2049:
2045:
2041:
2037:
2036:
2022:
2021:
2010:
2005:
2001:
1998:
1995:
1992:
1989:
1986:
1981:
1977:
1973:
1970:
1967:
1964:
1961:
1956:
1952:
1948:
1945:
1942:
1939:
1936:
1931:
1927:
1923:
1920:
1917:
1914:
1911:
1906:
1902:
1898:
1895:
1890:
1886:
1881:
1875:
1870:
1866:
1860:
1857:
1854:
1851:
1848:
1824:
1817:
1809:Leonhard Euler
1803:
1800:
1795:
1794:
1779:
1775:
1771:
1768:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1741:
1739:
1735:
1731:
1728:
1725:
1722:
1719:
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1708:
1704:
1700:
1694:
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1684:
1681:
1678:
1675:
1672:
1669:
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1644:
1637:
1632:
1624:
1621:
1616:
1612:
1608:
1605:
1600:
1597:
1594:
1591:
1588:
1585:
1580:
1576:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1547:
1543:
1540:
1537:
1535:
1533:
1530:
1527:
1524:
1521:
1520:
1498:
1497:
1486:
1482:
1479:
1476:
1471:
1467:
1463:
1460:
1457:
1454:
1449:
1443:
1439:
1435:
1432:
1428:
1424:
1421:
1418:
1415:
1412:
1409:
1376:
1373:
1312:
1311:
1300:
1296:
1293:
1287:
1284:
1279:
1275:
1268:
1264:
1260:
1256:
1253:
1246:
1241:
1237:
1233:
1230:
1227:
1224:
1221:
1218:
1200:
1199:
1184:
1180:
1177:
1169:
1166:
1161:
1156:
1152:
1149:
1144:
1140:
1134:
1130:
1126:
1123:
1119:
1112:
1107:
1103:
1099:
1094:
1090:
1086:
1083:
1080:
1077:
1074:
1071:
1069:
1067:
1064:
1061:
1057:
1054:
1051:
1048:
1043:
1038:
1034:
1030:
1027:
1025:
1023:
1020:
1017:
1014:
1011:
1010:
970:
969:
958:
950:
947:
941:
937:
934:
929:
925:
919:
915:
911:
908:
904:
898:
893:
889:
885:
882:
879:
876:
870:
867:
864:
861:
858:
839:
836:
835:
834:
819:
810:
806:
802:
799:
796:
793:
788:
785:
779:
774:
770:
765:
757:
753:
749:
746:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
712:
710:
707:
706:
703:
696:
693:
690:
686:
681:
675:
672:
669:
664:
661:
658:
652:
649:
645:
641:
638:
635:
632:
629:
626:
623:
618:
614:
609:
603:
599:
596:
593:
587:
584:
582:
580:
577:
576:
503:
500:
482:
479:
461:
458:
456: toises.
418:Paris Meridian
391:Anders Celsius
234:pendulum clock
213:
210:
184:to describe a
174:Main article:
171:
168:
113:
110:
94:astro-geodetic
26:
9:
6:
4:
3:
2:
8287:
8276:
8273:
8271:
8268:
8266:
8263:
8262:
8260:
8251:
8248:
8247:
8236:
8231:
8224:
8220:
8216:
8212:
8208:
8204:
8199:
8194:
8190:
8186:
8179:
8170:
8162:
8158:
8154:
8150:
8146:
8145:
8140:
8134:
8125:
8118:
8117:
8112:
8107:
8098:
8090:
8086:
8082:
8078:
8074:
8070:
8069:
8064:
8060:
8054:
8047:
8043:
8039:
8033:
8024:
8017:
8011:
8007:
8003:
7999:
7995:
7991:
7986:
7981:
7977:
7973:
7972:Astron. Nachr
7969:
7968:Bessel, F. W.
7963:
7961:
7959:
7951:
7946:
7940:
7934:
7927:
7926:
7919:
7913:
7909:
7905:
7903:
7896:
7888:
7884:
7880:
7876:
7872:
7869:(in German).
7868:
7864:
7860:
7859:Bessel, F. W.
7854:
7845:
7838:
7836:
7829:
7827:
7819:
7814:
7811:(in French).
7810:
7806:
7802:
7796:
7789:
7784:
7778:
7773:
7766:
7760:
7753:
7749:
7744:
7740:
7736:
7729:
7722:
7716:
7708:
7702:
7698:
7691:
7682:
7678:
7674:
7673:
7665:
7658:
7654:
7650:
7646:
7645:9781440088650
7642:
7638:
7634:
7628:
7624:
7620:
7619:
7614:
7608:
7606:
7604:
7596:
7592:
7588:
7586:
7579:
7565:. p. 307
7564:
7560:
7556:
7549:
7541:
7537:
7536:
7528:
7514:
7508:
7504:
7503:
7495:
7487:
7482:
7481:
7475:
7469:
7465:
7455:
7452:
7450:
7447:
7445:
7442:
7440:
7437:
7435:
7432:
7429:
7426:
7424:
7421:
7419:
7416:
7414:
7411:
7410:
7403:
7369:
7357:
7335:
7332:
7329:
7324:
7320:
7313:
7310:
7304:
7302:
7296:
7292:
7288:
7281:
7278:
7275:
7270:
7266:
7259:
7256:
7250:
7245:
7241:
7234:
7231:
7225:
7223:
7217:
7213:
7209:
7201:
7198:
7195:
7190:
7186:
7179:
7176:
7170:
7168:
7162:
7158:
7154:
7148:
7145:
7142:
7137:
7133:
7126:
7123:
7117:
7114:
7108:
7105:
7099:
7097:
7091:
7087:
7083:
7071:
7070:
7069:
7052:
7049:
7046:
7043:
7040:
7037:
7034:
7030:
7026:
7022:
7018:
7015:
7012:
7009:
7006:
7002:
6998:
6994:
6990:
6987:
6984:
6981:
6978:
6974:
6970:
6966:
6962:
6959:
6956:
6953:
6950:
6946:
6942:
6938:
6934:
6931:
6928:
6925:
6918:
6917:
6916:
6887:
6884:
6881:
6876:
6872:
6865:
6862:
6856:
6854:
6848:
6844:
6840:
6833:
6830:
6827:
6822:
6818:
6811:
6808:
6802:
6797:
6793:
6786:
6783:
6777:
6775:
6769:
6765:
6761:
6753:
6750:
6747:
6742:
6738:
6731:
6728:
6722:
6720:
6714:
6710:
6706:
6700:
6697:
6694:
6689:
6685:
6678:
6675:
6669:
6666:
6660:
6657:
6651:
6649:
6643:
6639:
6635:
6623:
6622:
6621:
6604:
6601:
6598:
6595:
6592:
6589:
6585:
6581:
6577:
6573:
6570:
6567:
6564:
6561:
6557:
6553:
6549:
6545:
6542:
6539:
6536:
6533:
6529:
6525:
6521:
6517:
6514:
6511:
6508:
6505:
6501:
6497:
6493:
6489:
6486:
6483:
6480:
6473:
6472:
6471:
6468:
6464:
6460:
6453:
6449:
6444:
6415:
6411:
6404:
6401:
6396:
6393:
6386:
6385:
6384:
6381:
6374:
6354:
6342:
6338:
6331:
6326:
6323:
6315:
6311:
6304:
6298:
6293:
6289:
6285:
6280:
6277:
6274:
6270:
6262:
6261:
6260:
6258:
6240:
6210:
6206:
6201:
6194:
6191:
6188:
6182:
6179:
6174:
6170:
6162:
6161:
6160:
6158:
6150:
6143:
6138:
6117:
6113:
6109:
6106:
6101:
6097:
6089:
6088:
6087:
6085:
6055:
6051:
6047:
6044:
6042:
6034:
6026:
6023:
6015:
6012:
6010:
5997:
5985:
5984:
5983:
5980:
5978:
5974:
5967:
5948:
5942:
5939:
5935:
5929:
5924:
5918:
5915:
5909:
5906:
5898:
5895:
5889:
5886:
5883:
5880:
5871:
5859:
5856:
5853:
5849:
5843:
5836:
5833:
5830:
5824:
5818:
5813:
5809:
5803:
5796:
5793:
5790:
5784:
5778:
5767:
5759:
5758:
5757:
5754:
5752:
5735:
5732:
5728:
5725:
5702:
5695:
5692:
5688:
5682:
5679:
5676:
5670:
5664:
5661:
5658:
5647:
5639:
5638:
5637:
5635:
5630:
5628:
5624:
5623:nautical mile
5620:
5601:
5596:
5591:
5588:
5583:
5579:
5576:
5565:
5557:
5556:
5555:
5553:
5549:
5545:
5536:
5531:
5516:
5512:
5509:
5501:
5495:
5491:
5487:
5483:
5479:
5476:
5469:
5468:main category
5465:
5464:
5459:
5456:
5452:
5448:
5445:
5442:
5441:
5435:
5429:
5427:
5422:You can help
5418:
5409:
5408:
5385: metres.
5376:
5373:
5370:
5367:
5364:
5361:
5358:
5354:
5348:
5345:
5335:
5334:
5333:
5326:
5315:
5311:
5304:
5300:
5291:
5282:
5276:
5272:
5252:
5244:
5219: metres,
5212:
5208:
5205:
5202:
5199:
5193:
5190:
5186:
5182:
5179:
5176:
5173:
5170:
5167:
5164:
5160:
5157:
5154:
5151:
5148:
5145:
5141:
5138:
5135:
5132:
5129:
5126:
5122:
5118:
5115:
5107:
5100:
5095:
5091:
5087:
5083:
5079:
5077:
5061:
5057:
5054:
5051:
5048:
5044:
5040:
5037:
5034:
5031:
5028:
5025:
5021:
5018:
5015:
5012:
5009:
5006:
5002:
4999:
4996:
4993:
4990:
4987:
4983:
4979:
4976:
4968:
4961:
4956:
4952:
4948:
4944:
4940:
4938:
4930:
4924:
4913:
4912:
4911:
4909:
4904:
4897:
4893:
4886:
4882:
4877:
4867:
4840:
4836:
4832:
4829:
4826:
4823:
4820:
4817:
4814:
4811:
4808:
4803:
4800:
4797:
4794:
4791:
4788:
4775:
4771:
4764:
4762:
4743:
4737:
4734:
4730:
4723:
4720:
4717:
4714:
4705:
4702:
4699:
4696:
4688:
4680:
4677:
4671:
4666:
4663:
4659:
4651:
4650:
4649:
4646:
4636:
4630:
4608:
4605:
4602:
4599:
4595:
4588:
4585:
4579:
4576:
4573:
4570:
4567:
4560:
4557:
4551:
4548:
4540:
4537:
4531:
4528:
4525:
4522:
4519:
4516:
4513:
4506:
4503:
4497:
4494:
4491:
4488:
4472:
4469:
4466:
4462:
4458:
4453:
4449:
4441:
4440:
4439:
4415:
4411:
4408:
4405:
4395:
4388:
4383:
4379:
4368:
4364:
4361:
4358:
4348:
4342:
4338:
4331:
4326:
4321:
4318:
4314:
4306:
4305:
4304:
4301:
4274:
4259:
4255:
4251:
4248:
4245:
4242:
4239:
4236:
4233:
4230:
4227:
4222:
4219:
4216:
4213:
4210:
4207:
4201:
4198:
4190:
4187:
4184:
4181:
4178:
4175:
4170:
4166:
4162:
4159:
4156:
4153:
4150:
4145:
4141:
4137:
4134:
4131:
4128:
4125:
4120:
4116:
4112:
4109:
4106:
4103:
4100:
4095:
4091:
4087:
4084:
4079:
4075:
4060:
4056:
4053:
4050:
4044:
4038:
4032:
4021:
4020:
4019:
4017:
3994:
3991:
3988:
3983:
3979:
3972:
3969:
3963:
3960:
3958:
3951:
3947:
3940:
3937:
3934:
3929:
3925:
3918:
3915:
3909:
3904:
3900:
3893:
3890:
3884:
3881:
3879:
3872:
3868:
3860:
3857:
3854:
3849:
3845:
3838:
3835:
3829:
3826:
3824:
3817:
3813:
3807:
3804:
3801:
3796:
3792:
3785:
3782:
3776:
3773:
3767:
3764:
3758:
3755:
3753:
3746:
3742:
3734:
3728:
3724:
3720:
3717:
3714:
3709:
3705:
3698:
3695:
3689:
3684:
3680:
3673:
3670:
3664:
3661:
3658:
3656:
3649:
3645:
3633:
3632:
3631:
3614:
3610:
3606:
3603:
3600:
3597:
3594:
3591:
3586:
3582:
3578:
3575:
3572:
3569:
3566:
3561:
3557:
3553:
3550:
3547:
3544:
3541:
3536:
3532:
3528:
3525:
3522:
3519:
3516:
3511:
3507:
3503:
3500:
3495:
3491:
3486:
3480:
3476:
3473:
3470:
3464:
3458:
3452:
3445:
3444:
3443:
3441:
3434:
3424:
3422:
3418:
3414:
3391:
3384:
3381:
3376:
3372:
3368:
3363:
3359:
3355:
3350:
3346:
3342:
3339:
3336:
3333:
3327:
3324:
3318:
3315:
3312:
3308:
3303:
3297:
3294:
3291:
3282:
3279:
3269:
3268:
3267:
3256:
3252:
3247:
3240:
3236:
3184:
3181:
3178:
3173:
3169:
3162:
3159:
3153:
3151:
3144:
3140:
3133:
3130:
3127:
3122:
3118:
3111:
3108:
3102:
3097:
3093:
3086:
3083:
3077:
3075:
3068:
3064:
3056:
3053:
3050:
3045:
3041:
3034:
3031:
3025:
3022:
3020:
3013:
3009:
3003:
3000:
2997:
2992:
2988:
2981:
2978:
2972:
2969:
2963:
2960:
2954:
2951:
2949:
2942:
2938:
2930:
2927:
2924:
2919:
2915:
2908:
2905:
2899:
2894:
2890:
2883:
2880:
2874:
2871:
2868:
2866:
2859:
2855:
2843:
2842:
2841:
2824:
2819:
2815:
2812:
2809:
2806:
2803:
2800:
2795:
2791:
2787:
2784:
2781:
2778:
2775:
2770:
2766:
2762:
2759:
2756:
2753:
2750:
2745:
2741:
2737:
2734:
2731:
2728:
2725:
2720:
2716:
2712:
2709:
2704:
2700:
2695:
2689:
2685:
2682:
2679:
2673:
2667:
2661:
2654:
2653:
2652:
2650:
2646:
2627:
2618:
2610:
2607:
2604:
2596:
2593:
2587:
2582:
2578:
2570:
2569:
2568:
2563:
2549:
2528:
2525:
2522:
2517:
2513:
2506:
2503:
2497:
2495:
2488:
2484:
2476:
2473:
2470:
2465:
2461:
2454:
2451:
2445:
2440:
2436:
2429:
2426:
2420:
2417:
2415:
2408:
2404:
2396:
2393:
2390:
2385:
2381:
2374:
2371:
2365:
2360:
2356:
2349:
2346:
2340:
2335:
2331:
2324:
2321:
2315:
2313:
2306:
2302:
2294:
2291:
2288:
2283:
2279:
2272:
2269:
2263:
2258:
2254:
2247:
2244:
2238:
2233:
2229:
2222:
2219:
2213:
2208:
2204:
2197:
2194:
2188:
2185:
2183:
2176:
2172:
2164:
2161:
2158:
2153:
2149:
2142:
2139:
2133:
2128:
2124:
2117:
2114:
2108:
2103:
2099:
2092:
2089:
2083:
2078:
2074:
2067:
2064:
2058:
2055:
2052:
2050:
2043:
2039:
2027:
2026:
2025:
2008:
2003:
1999:
1996:
1993:
1990:
1987:
1984:
1979:
1975:
1971:
1968:
1965:
1962:
1959:
1954:
1950:
1946:
1943:
1940:
1937:
1934:
1929:
1925:
1921:
1918:
1915:
1912:
1909:
1904:
1900:
1896:
1893:
1888:
1884:
1879:
1873:
1868:
1864:
1858:
1852:
1846:
1839:
1838:
1837:
1834:
1829:
1816:
1814:
1810:
1799:
1777:
1769:
1766:
1762:
1759:
1756:
1750:
1747:
1744:
1742:
1733:
1726:
1723:
1720:
1714:
1706:
1702:
1698:
1692:
1688:
1682:
1676:
1673:
1670:
1664:
1660:
1656:
1653:
1651:
1642:
1635:
1630:
1622:
1619:
1614:
1610:
1606:
1603:
1598:
1595:
1592:
1589:
1586:
1583:
1578:
1574:
1567:
1561:
1558:
1555:
1549:
1545:
1541:
1538:
1536:
1528:
1522:
1511:
1510:
1509:
1507:
1503:
1484:
1477:
1474:
1469:
1465:
1461:
1458:
1447:
1441:
1437:
1433:
1430:
1426:
1422:
1419:
1413:
1407:
1400:
1399:
1398:
1396:
1392:
1388:
1382:
1372:
1357:
1350:
1341:
1333:
1327:
1323:
1319:
1298:
1294:
1291:
1285:
1282:
1277:
1273:
1266:
1262:
1258:
1254:
1251:
1244:
1239:
1235:
1231:
1228:
1222:
1216:
1209:
1208:
1207:
1205:
1182:
1178:
1175:
1167:
1164:
1159:
1154:
1150:
1147:
1142:
1138:
1132:
1128:
1124:
1121:
1117:
1110:
1105:
1101:
1092:
1088:
1084:
1081:
1075:
1072:
1070:
1062:
1059:
1052:
1046:
1041:
1036:
1032:
1028:
1026:
1018:
1012:
1001:
1000:
999:
988:
984:
980:
976:
956:
948:
945:
939:
935:
932:
927:
923:
917:
913:
909:
906:
902:
891:
887:
883:
880:
874:
868:
862:
856:
849:
848:
847:
845:
817:
808:
800:
797:
794:
786:
783:
777:
772:
768:
763:
755:
751:
747:
744:
739:
736:
730:
727:
724:
718:
715:
713:
708:
701:
694:
691:
688:
684:
679:
673:
670:
667:
662:
659:
656:
650:
647:
643:
636:
633:
630:
624:
621:
616:
612:
607:
601:
597:
594:
591:
585:
583:
578:
567:
566:
565:
560:
552:
535:
533:
525:
524:
519:
515:
509:
499:
497:
493:
488:
487:nautical mile
478:
476:
472:
468:
457:
448:
433:
419:
415:
411:
407:
403:
398:
396:
392:
388:
384:
380:
376:
372:
368:
364:
360:
356:
352:
348:
344:
340:
336:
331:
327:
323:
303:
299:
295:
294:
289:
284:
282:
278:
274:
270:
265:
261:
243:
242:French Guiana
239:
235:
231:
227:
223:
219:
209:
207:
203:
199:
195:
191:
187:
183:
177:
167:
165:
161:
157:
153:
149:
145:
141:
137:
133:
129:
123:
119:
109:
107:
103:
99:
95:
91:
87:
82:
80:
76:
72:
68:
63:
61:
57:
53:
49:
45:
41:
37:
33:
19:
8191:(1): 43–55.
8188:
8184:
8178:
8169:
8148:
8143:
8133:
8124:
8115:
8106:
8097:
8072:
8066:
8053:
8045:
8032:
8023:
8015:
7975:
7971:
7945:
7933:
7923:
7918:
7901:
7895:
7870:
7866:
7853:
7844:
7834:
7812:
7808:
7795:
7783:
7772:
7759:
7748:Maxima files
7734:
7728:
7715:
7696:
7690:
7671:
7664:
7617:
7584:
7578:
7567:. Retrieved
7558:
7548:
7534:
7527:
7516:. Retrieved
7501:
7494:
7479:
7474:Russo, Lucio
7468:
7382:replaced by
7374:replaced by
7362:in terms of
7354:
7067:
6911:in terms of
6906:
6619:
6469:
6462:
6458:
6451:
6447:
6440:
6379:
6372:
6369:
6259:, iterating
6251:, determine
6246:
6227:
6148:
6141:
6134:
6081:
5981:
5965:
5963:
5755:
5717:
5631:
5627:hebdomometre
5616:
5551:
5543:
5541:
5490:edit summary
5481:
5461:
5431:
5423:
5332:is given by
5324:
5313:
5309:
5302:
5298:
5289:
5280:
5277:
5270:
5250:
5242:
5239:
5068: metres
4905:
4895:
4891:
4884:
4880:
4873:
4773:
4769:
4765:
4758:
4647:
4628:
4625:
4437:
4302:
4298:
4013:
3629:
3430:
3406:
3257:
3250:
3245:
3238:
3234:
3203:
2839:
2642:
2559:
2547:
2023:
1832:
1826:
1805:
1796:
1499:
1384:
1358:
1348:
1339:
1331:
1325:
1321:
1317:
1313:
1201:
986:
982:
978:
974:
971:
841:
551:eccentricity
536:
522:
521:
511:
496:day skippers
484:
463:
460:19th century
399:
387:Abbe Outhier
334:
328:and his son
291:
288:Isaac Newton
285:
279:than at the
264:acceleration
215:
205:
201:
192:in place of
189:
181:
179:
151:
144:Eratosthenes
125:
83:
78:
64:
58:, or to its
40:meridian arc
39:
29:
7633:Archive.org
6237:.146 m
5977:James Ivory
5621:and of the
4643:(−3)!! = −1
502:Calculation
471:Clarke 1866
467:Bessel 1841
351:Louis Godin
322:Jean Picard
226:Jean Richer
140:Alexandrian
8259:Categories
7815:: 258–293.
7649:Nabu Press
7569:2009-01-28
7518:2021-05-02
7461:References
6082:The polar
5528:See also:
5433:(May 2021)
4639:(−1)!! = 1
1393:handbook (
838:Definition
559:flattening
506:See also:
383:Le Monnier
371:Maupertuis
367:to Lapland
363:Jorge Juan
302:flattening
156:Posidonius
142:scientist
36:navigation
8223:119310141
8198:1109.4448
8161:312469983
8089:251572677
8059:Ivory, J.
8010:118760590
7985:0908.1824
7801:Euler, L.
7585:Principia
7333:⋯
7330:−
7279:⋯
7251:−
7199:⋯
7196:−
7146:⋯
7118:−
7050:⋯
7044:μ
7038:
7016:μ
7010:
6988:μ
6982:
6960:μ
6954:
6932:μ
6926:β
6885:⋯
6831:⋯
6803:−
6751:⋯
6698:⋯
6670:−
6605:⋯
6599:μ
6593:
6571:μ
6565:
6543:μ
6537:
6515:μ
6509:
6487:μ
6481:φ
6402:π
6394:μ
6339:φ
6324:−
6312:φ
6299:−
6290:φ
6271:φ
6137:perimeter
6024:π
5887:−
5865:∞
5850:∑
5825:π
5785:π
5589:π
5508:talk page
5460:Consider
5428:in German
5377:φ
5371:
5362:−
5343:Δ
5209:β
5203:
5191:−
5183:×
5177:−
5174:β
5168:
5158:−
5155:β
5149:
5139:−
5136:β
5130:
5116:−
5108:∘
5101:β
5058:φ
5052:
5035:φ
5029:
5019:−
5016:φ
5010:
4997:φ
4991:
4977:−
4969:∘
4962:φ
4931:φ
4830:φ
4824:
4804:φ
4798:
4700:−
4678:−
4529:−
4495:−
4478:∞
4463:∑
4249:φ
4243:
4223:φ
4217:
4202:−
4191:⋯
4188:−
4185:φ
4179:
4160:φ
4154:
4138:−
4135:φ
4129:
4110:φ
4104:
4088:−
4085:φ
4039:φ
3992:⋯
3964:−
3938:⋯
3885:−
3858:⋯
3830:−
3805:⋯
3759:−
3718:⋯
3607:⋯
3601:β
3595:
3576:β
3570:
3551:β
3545:
3526:β
3520:
3501:β
3459:φ
3440:geodesics
3385:⋯
3382:−
3356:−
3337:−
3182:⋯
3179:−
3131:⋯
3128:−
3103:−
3054:⋯
3026:−
3001:⋯
2955:−
2928:⋯
2816:⋯
2810:φ
2804:
2785:φ
2779:
2760:φ
2754:
2735:φ
2729:
2710:φ
2668:φ
2643:In 1837,
2526:⋯
2474:⋯
2471:−
2446:−
2421:−
2394:⋯
2292:⋯
2289:−
2264:−
2239:−
2214:−
2189:−
2162:⋯
2000:⋯
1994:φ
1988:
1969:φ
1963:
1944:φ
1938:
1919:φ
1913:
1894:φ
1853:φ
1815:squared.
1757:β
1721:φ
1703:φ
1671:φ
1636:φ
1623:
1607:−
1599:φ
1596:
1590:φ
1587:
1568:−
1556:φ
1529:φ
1459:φ
1453:Π
1434:−
1414:φ
1295:β
1286:β
1283:
1245:β
1236:∫
1223:φ
1179:φ
1160:−
1151:φ
1148:
1125:−
1111:φ
1102:∫
1085:−
1063:φ
1053:φ
1042:φ
1033:∫
1019:φ
936:φ
933:
910:−
884:−
863:φ
748:−
728:−
692:−
660:−
634:−
595:−
304:equal to
293:Principia
286:In 1687,
222:spherical
206:ellipsoid
198:ellipsoid
164:Al-Ma'mun
48:longitude
8141:(1811).
8113:(1921).
8061:(1798).
7861:(1837).
7803:(1755).
7615:(1880).
7595:pdf file
7476:(2004).
7406:See also
7297:′
7218:′
7163:′
7092:′
7031:′
7003:′
6975:′
6947:′
6849:′
6770:′
6715:′
6644:′
6586:′
6558:′
6530:′
6502:′
6061: m.
5736:′
5696:′
5484:provide
4402:if
4355:if
3419:and the
3255:vanish.
3204:Because
1828:Delambre
1770:′
1263:′
375:Clairaut
298:spheroid
269:latitude
202:Spheroid
194:spheroid
56:meridian
8265:Geodesy
8235:Addenda
8203:Bibcode
7990:Bibcode
7875:Bibcode
7818:Figures
7627:2484948
7618:Geodesy
7502:Geodesy
7418:Geodesy
6441:is the
6056:965.729
5506:to the
5488:in the
5430:.
5261:
5247:
5123:346.170
5092:132.952
4984:038.509
4953:132.952
4772:)(1 + 2
4633:is the
3415:by the
3231:
3219:
2649:Helmert
1353:
1336:
1320:= (1 −
449:bar as
443: m
408:to the
406:Dunkirk
347:Bouguer
335:prolate
318:
306:
281:Equator
253:⁄
238:Cayenne
230:gravity
136:Baghdad
54:of the
52:segment
42:is the
32:geodesy
8221:
8159:
8087:
8048:, 1–13
8008:
7788:Maxima
7703:
7681:906501
7679:
7655:
7643:
7625:
7509:
7068:where
6620:where
6383:where
6032:
6019:
5718:where
5240:where
5003:16.833
4768:(1 − 2
4438:where
2507:131072
2024:where
1314:where
990:(with
432:toises
430:
365:) and
186:sphere
148:stadia
128:caliph
102:WGS 84
60:length
8219:S2CID
8193:arXiv
8147:[
8085:S2CID
8006:S2CID
7980:arXiv
7750:and
7488:-277.
7370:with
5554:, is
5451:DeepL
5161:0.001
5142:1.122
5041:0.000
5022:0.022
4908:WGS84
3630:with
2840:with
2375:16384
2143:16384
2140:11025
1324:)tan
451:0.513
447:metre
412:(the
379:Camus
260:Paris
104:(see
75:geoid
44:curve
8157:OCLC
7701:ISBN
7677:OCLC
7653:ISBN
7641:ISBN
7635:and
7623:OCLC
7507:ISBN
7394:and
7314:1536
6863:1097
6159:is:
6147:= 2π
6135:The
5482:must
5480:You
5444:View
5308:) −
5287:and
5245:° =
4890:) −
4641:and
4626:and
4409:>
3212:and
2455:4096
2430:3072
2372:2205
2350:1024
2273:4096
2270:2205
2248:1024
1391:NIST
1347:1 −
1334:′ =
1329:and
1316:tan
842:The
453:0762
204:and
120:and
38:, a
34:and
8211:doi
8153:180
8077:doi
7998:doi
7976:331
7908:doi
7883:doi
7739:doi
7486:273
7311:539
7035:sin
7007:sin
6979:sin
6951:sin
6866:512
6729:151
6590:sin
6562:sin
6534:sin
6506:sin
6235:449
6232:367
6183:0.5
6052:001
5453:or
5368:cos
5365:560
5359:133
5355:111
5296:is
5275:).
5264:is
5200:sin
5180:0.5
5165:sin
5146:sin
5127:sin
5088:111
5049:sin
5026:sin
5007:sin
4988:sin
4949:111
4821:cos
4795:sin
4240:cos
4214:sin
4176:sin
4151:sin
4126:sin
4101:sin
4018:as
3973:512
3592:sin
3567:sin
3542:sin
3517:sin
3262:or
3163:512
3160:315
2801:sin
2776:sin
2751:sin
2726:sin
2504:315
2452:105
2347:105
2325:256
2245:525
2118:256
2115:175
1985:sin
1960:sin
1935:sin
1910:sin
1620:sin
1593:cos
1584:sin
1508:),
1397:),
1274:sin
1139:sin
998:is
924:sin
441:000
438:000
427:762
424:130
315:230
300:of
236:to
134:in
130:'s
108:).
30:In
8261::
8217:.
8209:.
8201:.
8189:87
8187:.
8155:.
8083:.
8071:.
8065:.
8046:59
8044:,
8004:.
7996:.
7988:.
7974:.
7957:^
7881:.
7871:14
7825:^
7737:,
7602:^
7557:.
7540:20
7402:.
7260:96
7257:37
7235:16
7180:96
7177:29
7127:32
6812:32
6809:55
6787:16
6784:21
6732:96
6679:32
6676:27
6378:=
6239:.
6048:10
5979:.
5753:.
5636:,
5629:.
5258:1°
5187:10
5096:55
5045:03
4980:16
4957:55
4866:.
4645:.
4631:!!
3919:64
3894:16
3839:48
3786:16
3699:64
3442:,
3423:.
3237:+
3112:64
3109:15
3087:16
3084:15
3035:48
3032:35
2982:16
2909:64
2651:,
2427:35
2322:15
2223:32
2220:15
2093:64
2090:45
1836:,
1363:,
1356:.
1206:,
987:dφ
985:)
977:=
975:dm
553:,
545:,
541:,
534:.
477:.
436:10
389:,
385:,
381:,
377:,
373:,
361:,
357:,
353:,
349:,
283:.
271:,
240:,
166:.
62:.
8237:.
8225:.
8213::
8205::
8195::
8163:.
8091:.
8079::
8073:4
8016:4
8012:.
8000::
7992::
7982::
7910::
7889:.
7885::
7877::
7820:.
7813:9
7754:)
7741::
7709:.
7683:.
7651:(
7639:(
7629:.
7572:.
7542:.
7521:.
7400:μ
7396:τ
7392:n
7388:ε
7384:σ
7380:β
7376:s
7372:m
7364:β
7360:m
7336:.
7325:4
7321:n
7305:=
7293:8
7289:B
7282:,
7276:+
7271:4
7267:n
7246:2
7242:n
7232:5
7226:=
7214:4
7210:B
7202:,
7191:3
7187:n
7171:=
7159:6
7155:B
7149:,
7143:+
7138:3
7134:n
7124:9
7115:n
7109:2
7106:1
7100:=
7088:2
7084:B
7053:,
7047:+
7041:8
7027:8
7023:B
7019:+
7013:6
6999:6
6995:B
6991:+
6985:4
6971:4
6967:B
6963:+
6957:2
6943:2
6939:B
6935:+
6929:=
6913:β
6909:m
6888:.
6882:+
6877:4
6873:n
6857:=
6845:8
6841:H
6834:,
6828:+
6823:4
6819:n
6798:2
6794:n
6778:=
6766:4
6762:H
6754:,
6748:+
6743:3
6739:n
6723:=
6711:6
6707:H
6701:,
6695:+
6690:3
6686:n
6667:n
6661:2
6658:3
6652:=
6640:2
6636:H
6602:+
6596:8
6582:8
6578:H
6574:+
6568:6
6554:6
6550:H
6546:+
6540:4
6526:4
6522:H
6518:+
6512:2
6498:2
6494:H
6490:+
6484:=
6465:)
6463:φ
6461:(
6459:M
6454:)
6452:φ
6450:(
6448:m
6421:p
6416:m
6412:m
6405:2
6397:=
6380:μ
6376:0
6373:φ
6355:,
6348:)
6343:i
6335:(
6332:M
6327:m
6321:)
6316:i
6308:(
6305:m
6294:i
6286:=
6281:1
6278:+
6275:i
6253:φ
6249:m
6230:6
6211:0
6207:c
6202:/
6198:)
6195:b
6192:+
6189:a
6186:(
6180:=
6175:r
6171:M
6152:r
6149:M
6145:p
6142:C
6118:p
6114:m
6110:4
6107:=
6102:p
6098:C
6045:=
6035:a
6027:2
6013:=
6003:p
5998:m
5969:0
5966:c
5949:,
5943:j
5940:2
5936:n
5930:2
5925:)
5919:!
5916:!
5913:)
5910:j
5907:2
5904:(
5899:!
5896:!
5893:)
5890:3
5884:j
5881:2
5878:(
5872:(
5860:0
5857:=
5854:j
5844:4
5840:)
5837:b
5834:+
5831:a
5828:(
5819:=
5814:0
5810:c
5804:4
5800:)
5797:b
5794:+
5791:a
5788:(
5779:=
5773:p
5768:m
5733:e
5729:,
5726:e
5703:.
5700:)
5693:e
5689:i
5686:(
5683:E
5680:b
5677:=
5674:)
5671:e
5668:(
5665:E
5662:a
5659:=
5653:p
5648:m
5602:.
5597:)
5592:2
5584:(
5580:m
5577:=
5571:p
5566:m
5517:.
5510:.
5380:)
5374:2
5352:(
5349:=
5346:m
5330:φ
5325:m
5323:Δ
5319:)
5317:2
5314:φ
5312:(
5310:m
5306:1
5303:φ
5301:(
5299:m
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5290:φ
5284:1
5281:φ
5273:°
5271:β
5266:φ
5255:/
5251:φ
5243:φ
5213:)
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5194:6
5171:6
5152:4
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5111:)
5105:(
5084:(
5080:=
5062:)
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5038:+
5032:6
5013:4
5000:+
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4966:(
4945:(
4941:=
4934:)
4928:(
4925:m
4901:)
4899:2
4896:φ
4894:(
4892:m
4888:1
4885:φ
4883:(
4881:m
4864:β
4860:φ
4841:2
4837:n
4833:+
4827:2
4818:n
4815:2
4812:+
4809:1
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4792:n
4789:2
4776:)
4774:k
4770:k
4744:.
4738:k
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4731:B
4727:)
4724:k
4721:2
4718:+
4715:1
4712:(
4709:)
4706:k
4703:2
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4694:(
4689:k
4685:)
4681:1
4675:(
4672:=
4667:k
4664:2
4660:H
4629:k
4609:j
4606:2
4603:+
4600:k
4596:n
4589:!
4586:!
4583:)
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4577:2
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4565:(
4561:!
4558:!
4555:)
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4541:!
4538:!
4535:)
4532:3
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4517:j
4514:2
4511:(
4507:!
4504:!
4501:)
4498:3
4492:j
4489:2
4486:(
4473:0
4470:=
4467:j
4459:=
4454:k
4450:c
4416:,
4412:0
4406:k
4396:,
4389:k
4384:k
4380:c
4369:,
4365:0
4362:=
4359:k
4349:,
4343:0
4339:c
4332:{
4327:=
4322:k
4319:2
4315:B
4275:.
4270:)
4260:2
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4220:2
4211:n
4208:2
4182:8
4171:8
4167:B
4163:+
4157:6
4146:6
4142:B
4132:4
4121:4
4117:B
4113:+
4107:2
4096:2
4092:B
4080:0
4076:B
4068:(
4061:2
4057:b
4054:+
4051:a
4045:=
4042:)
4036:(
4033:m
3995:.
3989:+
3984:4
3980:n
3970:5
3961:=
3952:8
3948:B
3941:,
3935:+
3930:4
3926:n
3916:1
3910:+
3905:2
3901:n
3891:1
3882:=
3873:4
3869:B
3861:,
3855:+
3850:3
3846:n
3836:1
3827:=
3818:6
3814:B
3808:,
3802:+
3797:3
3793:n
3783:1
3777:+
3774:n
3768:2
3765:1
3756:=
3747:2
3743:B
3735:,
3729:0
3725:H
3721:=
3715:+
3710:4
3706:n
3696:1
3690:+
3685:2
3681:n
3674:4
3671:1
3665:+
3662:1
3659:=
3650:0
3646:B
3615:,
3611:)
3604:+
3598:8
3587:8
3583:B
3579:+
3573:6
3562:6
3558:B
3554:+
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3537:4
3533:B
3529:+
3523:2
3512:2
3508:B
3504:+
3496:0
3492:B
3487:(
3481:2
3477:b
3474:+
3471:a
3465:=
3462:)
3456:(
3453:m
3436:β
3409:n
3392:,
3388:)
3377:4
3373:n
3369:+
3364:3
3360:n
3351:2
3347:n
3343:+
3340:n
3334:1
3331:(
3328:a
3325:=
3319:n
3316:+
3313:1
3309:a
3304:=
3301:)
3298:b
3295:+
3292:a
3289:(
3283:2
3280:1
3264:b
3260:a
3251:k
3249:2
3246:H
3241:)
3239:b
3235:a
3233:(
3228:2
3225:/
3222:1
3214:b
3210:a
3206:n
3185:.
3174:4
3170:n
3154:=
3145:8
3141:H
3134:,
3123:4
3119:n
3098:2
3094:n
3078:=
3069:4
3065:H
3057:,
3051:+
3046:3
3042:n
3023:=
3014:6
3010:H
3004:,
2998:+
2993:3
2989:n
2979:3
2973:+
2970:n
2964:2
2961:3
2952:=
2943:2
2939:H
2931:,
2925:+
2920:4
2916:n
2906:1
2900:+
2895:2
2891:n
2884:4
2881:1
2875:+
2872:1
2869:=
2860:0
2856:H
2825:,
2820:)
2813:+
2807:8
2796:8
2792:H
2788:+
2782:6
2771:6
2767:H
2763:+
2757:4
2746:4
2742:H
2738:+
2732:2
2721:2
2717:H
2713:+
2705:0
2701:H
2696:(
2690:2
2686:b
2683:+
2680:a
2674:=
2671:)
2665:(
2662:m
2628:.
2619:2
2615:)
2611:n
2608:+
2605:1
2602:(
2597:n
2594:4
2588:=
2583:2
2579:e
2565:n
2556:)
2554:n
2529:.
2523:+
2518:8
2514:e
2498:=
2489:8
2485:D
2477:,
2466:8
2462:e
2441:6
2437:e
2418:=
2409:6
2405:D
2397:,
2391:+
2386:8
2382:e
2366:+
2361:6
2357:e
2341:+
2336:4
2332:e
2316:=
2307:4
2303:D
2295:,
2284:8
2280:e
2259:6
2255:e
2234:4
2230:e
2209:2
2205:e
2198:8
2195:3
2186:=
2177:2
2173:D
2165:,
2159:+
2154:8
2150:e
2134:+
2129:6
2125:e
2109:+
2104:4
2100:e
2084:+
2079:2
2075:e
2068:4
2065:3
2059:+
2056:1
2053:=
2044:0
2040:D
2009:,
2004:)
1997:+
1991:8
1980:8
1976:D
1972:+
1966:6
1955:6
1951:D
1947:+
1941:4
1930:4
1926:D
1922:+
1916:2
1905:2
1901:D
1897:+
1889:0
1885:D
1880:(
1874:a
1869:2
1865:b
1859:=
1856:)
1850:(
1847:m
1833:e
1823:)
1821:e
1778:.
1774:)
1767:e
1763:i
1760:,
1754:(
1751:E
1748:b
1745:=
1734:)
1730:)
1727:e
1724:,
1718:(
1715:E
1707:2
1699:d
1693:2
1689:d
1683:+
1680:)
1677:e
1674:,
1668:(
1665:E
1661:(
1657:a
1654:=
1643:)
1631:2
1615:2
1611:e
1604:1
1579:2
1575:e
1565:)
1562:e
1559:,
1553:(
1550:E
1546:(
1542:a
1539:=
1532:)
1526:(
1523:m
1485:.
1481:)
1478:e
1475:,
1470:2
1466:e
1462:,
1456:(
1448:)
1442:2
1438:e
1431:1
1427:(
1423:a
1420:=
1417:)
1411:(
1408:m
1369:μ
1365:β
1361:φ
1349:e
1344:/
1340:e
1332:e
1326:φ
1322:f
1318:β
1299:,
1292:d
1278:2
1267:2
1259:e
1255:+
1252:1
1240:0
1232:b
1229:=
1226:)
1220:(
1217:m
1183:.
1176:d
1168:2
1165:3
1155:)
1143:2
1133:2
1129:e
1122:1
1118:(
1106:0
1098:)
1093:2
1089:e
1082:1
1079:(
1076:a
1073:=
1060:d
1056:)
1050:(
1047:M
1037:0
1029:=
1022:)
1016:(
1013:m
996:φ
992:φ
983:φ
981:(
979:M
957:,
949:2
946:3
940:)
928:2
918:2
914:e
907:1
903:(
897:)
892:2
888:e
881:1
878:(
875:a
869:=
866:)
860:(
857:M
818:.
809:2
805:)
801:n
798:+
795:1
792:(
787:n
784:4
778:=
773:2
769:e
764:,
756:2
752:e
745:1
740:a
737:=
734:)
731:f
725:1
722:(
719:a
716:=
709:b
702:,
695:f
689:2
685:f
680:=
674:b
671:+
668:a
663:b
657:a
651:=
648:n
644:,
640:)
637:f
631:2
628:(
625:f
622:=
617:2
613:e
608:,
602:a
598:b
592:a
586:=
579:f
562:n
555:e
547:f
543:b
539:a
528:φ
422:5
369:(
345:(
312:/
309:1
255:2
251:1
248:+
246:2
20:)
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