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# Great

[greyt]
Schism, Great, or Schism of the West, division in the Roman Catholic Church from 1378 to 1417. There was no question of faith or practice involved; the schism was a matter of persons and politics. Shortly after Gregory XI had returned the papacy from Avignon to Rome, he died (Mar. 27, 1378). The Romans feared that the papal court might be returned to Avignon, and there was rioting, with the mob demanding a Roman, or at least an Italian, pope. On Apr. 8 the 16 cardinals present elected Urban VI. The new pope was soon acting very offensively to all in the church. The cardinals met at Agnani and on Aug. 2 declared Urban's election null. At Fondi on Sept. 20 they elected Robert of Geneva pope as Clement VII. Urban VI remained in Rome, refusing to step down, and Clement VII fled to Avignon, where he reigned surrounded by the former Roman court. There were thus two lines of popes. The popes at Rome were Urban VI (1378-89), Boniface IX (1389-1404), Innocent VII (1404-6), and Gregory XII (1406-15). Those of the rival line at Avignon were Clement VII (1378-94) and Benedict XIII (1394-1417; see Luna, Pedro de). Schism within schism ensued. France withdrew from obedience to Benedict XIII and recognized no pope (1398-1403, 1408-9). Theologians of the Univ. of Paris, led by Pierre d'Ailly and John Gerson, were anxious to end the schism, and they developed the theory that popes are subject to general councils. The Council of Pisa (1409; see Pisa, Council of) was the result. This meeting declared that Gregory XII of the Roman (or Urbanist) line and Benedict XIII of the Avignon (or Clementine) line were not popes and elected another, Alexander V. He died soon after, but his energetic successor, Baldassare Cossa (John XXIII, 1410-15), detached most of Europe from his rivals. In 1414 John reluctantly convened the Council of Constance (see Constance, Council of). Gregory XII resigned. John XXIII and Benedict XIII, who refused to resign, were declared deposed by the council. Martin V was elected, and the schism was at an end. The main effects of the schism were to delay needed reforms in the church and to give rise to the conciliar theory, which was revived at the Council of Basel (see Basel, Council of). It is generally agreed by Roman Catholic scholars that the line of popes from Urban to Gregory was the canonical one.

See W. Ullmann, Origins of the Great Schism (1948, repr. 1972); B. Tierney, Foundations of the Conciliar Theory (1955, repr. 1969); E. F. Jacob, Essays in the Conciliar Epoch (3d ed. 1963); M. Gail, The Three Popes (1969); J. H. Smith, The Great Schism (1970).

Schütt, Great, Slovak Velký Žitný Ostrov, island, c.725 sq mi (1,880 sq km), SW Slovakia, in the Danubian lowlands between the Danube River and its northern arm. It extends c.55 mi (90 km) from Bratislava to Komárno. The island's fertile soil produces a variety of crops. Opposite the Great Schütt lies Szigetköz, or the Little Schütt, an island c.30 mi (50 km) long and up to c.10 mi (16 km) wide, in NW Hungary between the Danube River and its southern arm. Wheat, rye, and dairy products are produced there. In 1954 a disastrous flood submerged much of the island.
Belt, Great, and Little Belt, straits: see Store Bælt and Lille Bælt, straits, Denmark.
Seal of the United States, Great: see United States, Great Seal of the.
Trek, Great, the journey by Afrikaner farmers (Boers) who left the Cape Colony to escape British domination and eventually founded Natal, Transvaal, and the Orange Free State. Trek is an Afrikaans term, originally meaning a journey by ox wagon. In this most famous trek, 12,000 Boers left the Cape between 1835 and 1843. The Voortrekkers (as these Boers are known) migrated beyond the Orange River. After defeating resident Africans, most remained in the highveld of the interior, forming isolated communities and small states.

See E. Walker, The Great Trek (5th ed. 1965).

Khingan, Great, mountain range, China: see Hinggan Ling, Da.
The great-circle distance is the shortest distance between any two points on the surface of a sphere measured along a path on the surface of the sphere (as opposed to going through the sphere's interior). Because spherical geometry is rather different from ordinary Euclidean geometry, the equations for distance take on a different form. The distance between two points in Euclidean space is the length of a straight line from one point to the other. On the sphere, however, there are no straight lines. In non-Euclidean geometry, straight lines are replaced with Geodesics. Geodesics on the sphere are the great circles (circles on the sphere whose centers are coincident with the center of the sphere).

Between any two points on a sphere which are not directly opposite each other, there is a unique great circle. The two points separate the great circle into two arcs. The length of the shorter arc is the great-circle distance between the points. A great circle endowed with such a distance is the Riemannian circle.

Between two points which are directly opposite each other, called antipodal points, there are infinitely many great circles, but all great circle arcs between antipodal points have the same length, i.e. half the circumference of the circle, or $pi r$, where r is the radius of the sphere.

Because the Earth is approximately spherical (see Earth radius), the equations for great-circle distance are important for finding the shortest distance between points on the surface of the Earth, and so have important applications in navigation.

## The geographical formula

Let $phi_s,lambda_s; phi_f,lambda_f;!$ be the geographical latitude and longitude of two points (a base "standpoint" and the destination "forepoint"), respectively, $Deltalambda;!$ the longitude difference and $Deltawidehat\left\{sigma\right\};!$ the (spherical) angular difference/distance, or central angle, which can be constituted from the spherical law of cosines:

$\left\{color\left\{white\right\}Big|\right\}Deltawidehat\left\{sigma\right\}=arccosbig\left(cosphi_scosphi_fcosDeltalambda+sinphi_ssinphi_fbig\right).;!$

The distance d, i.e. the arc length, for a sphere of radius r and $Deltawidehat\left\{sigma\right\}!$ given in radians, is then:

$d = r Deltawidehat\left\{sigma\right\}.$

This arccosine formula above can have large rounding errors for the common case where the distance is small, however, so it is not normally used. Instead, an equation known historically as the haversine formula was preferred, which is much more accurate for small distances:

$\left\{color\left\{white\right\}frac\left\{bigg|\right\}$ =2arcsinleft(sqrt{sin^2left(frac{phi_f-phi_s}{2}right)+cos{phi_s}cos{phi_f}sin^2left(frac{Deltalambda}{2}right)}right).;!

(Historically, the use of this formula was simplified by the availability of tables for the haversine function: hav(θ) = sin2(θ/2).)

Although this formula is accurate for most distances, it too suffers from rounding errors for the special (and somewhat unusual) case of antipodal points (on opposite ends of the sphere). A more complicated formula that is accurate for all distances is the Vincenty formula:

$\left\{color\left\{white\right\}frac\left\{bigg$
>Deltawidehat{sigma}=arctanleft(frac{sqrt{left(cosphi_fsinDeltalambdaright)^2+left(cosphi_ssinphi_f-sinphi_scosphi_fcosDeltalambdaright)^2}}{sinphi_ssinphi_f+cosphi_scosphi_fcosDeltalambda}right);;!

(When programming a computer, one should use the `atan2()` function rather than the ordinary arctangent function (`atan()`), in order to simplify handling of the case where the denominator is zero.)

If r is the great-circle radius of the sphere, then the great-circle distance is $r,Deltawidehat\left\{sigma\right\};!$.

Note: above, accuracy refers to rounding errors only; all formulas themselves are exact (for a sphere).

## Spherical distance on the Earth

The shape of the Earth more closely resembles a flattened spheroid with extreme values for the radius of arc, or arcradius—the radius of curvature, of 6335.437 km at the equator (vertically) and 6399.592 km at the poles, and having an average great-circle radius of 6372.795 km (3438.461 nautical miles). (Note that "arcradius" or "radius of curvature" is not the distance from the center of the earth to the surface. The distance from the center to the surface is smaller at the poles than at the equator; the arcradius is larger at the poles than at the equator.) Using a sphere with a radius of 6372.795 km thus results in an error of up to about 0.5%.

## A worked example

In order to use this formula for anything practical you will need two sets of coordinates. For example, the latitude and longitude of two airports:

• Nashville International Airport (BNA) in Nashville, TN, USA: N 36°7.2', W 86°40.2'
• Los Angeles International Airport (LAX) in Los Angeles, CA, USA: N 33°56.4', W 118°24.0'

First, convert these coordinates to decimal degrees (Sign × (Deg + (Min + Sec / 60) / 60)) and radians (× π / 180) before you can use them effectively in a formula. After conversion, the coordinates become:

• BNA: $phi_s= 36.12^circapprox 0.6304mbox\left\{ rad\right\};;;lambda_s=-86.67^circapprox -1.5127mbox\left\{ rad\right\};;!$
• LAX: $phi_f= 33.94^circapprox 0.5924mbox\left\{ rad\right\};;;lambda_f=-118.40^circapprox -2.0665mbox\left\{ rad\right\};;!$

Using these values in the angular difference/distance equation:

$r,Deltawidehat\left\{sigma\right\}approx 6372.795times0.45306 approx 2887.259mbox\left\{ km\right\}.;!$

Thus the distance between LAX and BNA is about 2887 km or 1794 miles (× 0.62137) or 1558 nautical miles (× 0.539553).