7805:
7207:
3838:
3203:
7282:
6696:
5612:
3208:
4913:
8680:
case, which is discussed below. For three given elements there are six cases: three sides, two sides and an included or opposite angle, two angles and an included or opposite side, or three angles. (The last case has no analogue in planar trigonometry.) No single method solves all cases. The figure below shows the seven non-trivial cases: in each case the given sides are marked with a cross-bar and the given angles with an arc. (The given elements are also listed below the triangle). In the summary notation here such as ASA, A refers to a given angle and S refers to a given side, and the sequence of A's and S's in the notation refers to the corresponding sequence in the triangle.
6185:
2930:
5069:
4283:
7800:{\displaystyle {\begin{alignedat}{4}&{\text{(Q1)}}&\qquad \cos C&=-\cos A\,\cos B,&\qquad \qquad &{\text{(Q6)}}&\qquad \tan B&=-\cos a\,\tan C,\\&{\text{(Q2)}}&\sin A&=\sin a\,\sin C,&&{\text{(Q7)}}&\tan A&=-\cos b\,\tan C,\\&{\text{(Q3)}}&\sin B&=\sin b\,\sin C,&&{\text{(Q8)}}&\cos a&=\sin b\,\cos A,\\&{\text{(Q4)}}&\tan A&=\tan a\,\sin B,&&{\text{(Q9)}}&\cos b&=\sin a\,\cos B,\\&{\text{(Q5)}}&\tan B&=\tan b\,\sin A,&&{\text{(Q10)}}&\cos C&=-\cot a\,\cot b.\end{alignedat}}}
7202:{\displaystyle {\begin{alignedat}{4}&{\text{(R1)}}&\qquad \cos c&=\cos a\,\cos b,&\qquad \qquad &{\text{(R6)}}&\qquad \tan b&=\cos A\,\tan c,\\&{\text{(R2)}}&\sin a&=\sin A\,\sin c,&&{\text{(R7)}}&\tan a&=\cos B\,\tan c,\\&{\text{(R3)}}&\sin b&=\sin B\,\sin c,&&{\text{(R8)}}&\cos A&=\sin B\,\cos a,\\&{\text{(R4)}}&\tan a&=\tan A\,\sin b,&&{\text{(R9)}}&\cos B&=\sin A\,\cos b,\\&{\text{(R5)}}&\tan b&=\tan B\,\sin a,&&{\text{(R10)}}&\cos c&=\cot A\,\cot B.\end{alignedat}}}
3833:{\displaystyle {\begin{alignedat}{5}{\text{(CT1)}}&&\qquad \cos b\,\cos C&=\cot a\,\sin b-\cot A\,\sin C\qquad &&(aCbA)\\{\text{(CT2)}}&&\cos b\,\cos A&=\cot c\,\sin b-\cot C\,\sin A&&(CbAc)\\{\text{(CT3)}}&&\cos c\,\cos A&=\cot b\,\sin c-\cot B\,\sin A&&(bAcB)\\{\text{(CT4)}}&&\cos c\,\cos B&=\cot a\,\sin c-\cot A\,\sin B&&(AcBa)\\{\text{(CT5)}}&&\cos a\,\cos B&=\cot c\,\sin a-\cot C\,\sin B&&(cBaC)\\{\text{(CT6)}}&&\cos a\,\cos C&=\cot b\,\sin a-\cot B\,\sin C&&(BaCb)\end{alignedat}}}
8818:
7217:
123:
5627:
6384:
3198:{\displaystyle \cos \!{\Bigl (}{\begin{smallmatrix}{\text{inner}}\\{\text{side}}\end{smallmatrix}}{\Bigr )}\cos \!{\Bigl (}{\begin{smallmatrix}{\text{inner}}\\{\text{angle}}\end{smallmatrix}}{\Bigr )}=\cot \!{\Bigl (}{\begin{smallmatrix}{\text{outer}}\\{\text{side}}\end{smallmatrix}}{\Bigr )}\sin \!{\Bigl (}{\begin{smallmatrix}{\text{inner}}\\{\text{side}}\end{smallmatrix}}{\Bigr )}-\cot \!{\Bigl (}{\begin{smallmatrix}{\text{outer}}\\{\text{angle}}\end{smallmatrix}}{\Bigr )}\sin \!{\Bigl (}{\begin{smallmatrix}{\text{inner}}\\{\text{angle}}\end{smallmatrix}}{\Bigr )},}
8684:
438:
207:
5607:{\displaystyle {\begin{aligned}{\frac {\sin {\tfrac {1}{2}}(A+B)}{\cos {\tfrac {1}{2}}C}}={\frac {\cos {\tfrac {1}{2}}(a-b)}{\cos {\tfrac {1}{2}}c}}&\qquad \qquad &{\frac {\sin {\tfrac {1}{2}}(A-B)}{\cos {\tfrac {1}{2}}C}}={\frac {\sin {\tfrac {1}{2}}(a-b)}{\sin {\tfrac {1}{2}}c}}\\{\frac {\cos {\tfrac {1}{2}}(A+B)}{\sin {\tfrac {1}{2}}C}}={\frac {\cos {\tfrac {1}{2}}(a+b)}{\cos {\tfrac {1}{2}}c}}&\qquad &{\frac {\cos {\tfrac {1}{2}}(A-B)}{\sin {\tfrac {1}{2}}C}}={\frac {\sin {\tfrac {1}{2}}(a+b)}{\sin {\tfrac {1}{2}}c}}\end{aligned}}}
4908:{\displaystyle {\begin{alignedat}{5}\sin {\tfrac {1}{2}}A&={\sqrt {\frac {\sin(s-b)\sin(s-c)}{\sin b\sin c}}}&\qquad \qquad \sin {\tfrac {1}{2}}a&={\sqrt {\frac {-\cos S\cos(S-A)}{\sin B\sin C}}}\\\cos {\tfrac {1}{2}}A&={\sqrt {\frac {\sin s\sin(s-a)}{\sin b\sin c}}}&\cos {\tfrac {1}{2}}a&={\sqrt {\frac {\cos(S-B)\cos(S-C)}{\sin B\sin C}}}\\\tan {\tfrac {1}{2}}A&={\sqrt {\frac {\sin(s-b)\sin(s-c)}{\sin s\sin(s-a)}}}&\tan {\tfrac {1}{2}}a&={\sqrt {\frac {-\cos S\cos(S-A)}{\cos(S-B)\cos(S-C)}}}\end{alignedat}}}
1243:
10856:"One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth"
6180:{\displaystyle {\begin{aligned}\tan {\tfrac {1}{2}}(A+B)={\frac {\cos {\tfrac {1}{2}}(a-b)}{\cos {\tfrac {1}{2}}(a+b)}}\cot {\tfrac {1}{2}}C&\qquad &\tan {\tfrac {1}{2}}(a+b)={\frac {\cos {\tfrac {1}{2}}(A-B)}{\cos {\tfrac {1}{2}}(A+B)}}\tan {\tfrac {1}{2}}c\\\tan {\tfrac {1}{2}}(A-B)={\frac {\sin {\tfrac {1}{2}}(a-b)}{\sin {\tfrac {1}{2}}(a+b)}}\cot {\tfrac {1}{2}}C&\qquad &\tan {\tfrac {1}{2}}(a-b)={\frac {\sin {\tfrac {1}{2}}(A-B)}{\sin {\tfrac {1}{2}}(A+B)}}\tan {\tfrac {1}{2}}c\end{aligned}}}
2425:
180:βare bounded by two great-circle arcs: a familiar example is the curved outward-facing surface of a segment of an orange. Three arcs serve to define a spherical triangle, the principal subject of this article. Polygons with higher numbers of sides (4-sided spherical quadrilaterals, 5-sided spherical pentagons, etc.) are defined in similar manner. Analogously to their plane counterparts, spherical polygons with more than 3 sides can always be treated as the composition of spherical triangles.
8952:
1994:
33:
4163:
2420:{\displaystyle {\begin{aligned}\sin ^{2}A&=1-\left({\frac {\cos a-\cos b\cos c}{\sin b\sin c}}\right)^{2}\\&={\frac {(1-\cos ^{2}b)(1-\cos ^{2}c)-(\cos a-\cos b\cos c)^{2}}{\sin ^{2}\!b\,\sin ^{2}\!c}}\\{\frac {\sin A}{\sin a}}&={\frac {\sqrt {1-\cos ^{2}\!a-\cos ^{2}\!b-\cos ^{2}\!c+2\cos a\cos b\cos c}}{\sin a\sin b\sin c}}.\end{aligned}}}
702:
8059:
2882:
6691:
10234:
1615:
963:
3850:
2456:
The derivation of the cosine rule presented above has the merits of simplicity and directness and the derivation of the sine rule emphasises the fact that no separate proof is required other than the cosine rule. However, the above geometry may be used to give an independent proof of the sine rule.
2452:
gives four different proofs of the cosine rule. Text books on geodesy and spherical astronomy give different proofs and the online resources of MathWorld provide yet more. There are even more exotic derivations, such as that of
Banerjee who derives the formulae using the linear algebra of projection
711:
then we can immediately derive a second identity by applying the first identity to the polar triangle by making the above substitutions. This is how the supplemental cosine equations are derived from the cosine equations. Similarly, the identities for a quadrantal triangle can be derived from those
8798:
The solution methods listed here are not the only possible choices: many others are possible. In general it is better to choose methods that avoid taking an inverse sine because of the possible ambiguity between an angle and its supplement. The use of half-angle formulae is often advisable because
8679:
The solution of triangles is the principal purpose of spherical trigonometry: given three, four or five elements of the triangle, determine the others. The case of five given elements is trivial, requiring only a single application of the sine rule. For four given elements there is one non-trivial
9501:
7817:
2635:
9753:
6483:
6374:
10039:
9107:
1392:
6475:
The key for remembering which trigonometric function goes with which part is to look at the first vowel of the kind of part: middle parts take the sine, adjacent parts take the tangent, and opposite parts take the cosine. For an example, starting with the sector containing
2447:
There are many ways of deriving the fundamental cosine and sine rules and the other rules developed in the following sections. For example, Todhunter gives two proofs of the cosine rule (Articles 37 and 60) and two proofs of the sine rule (Articles 40 and 42). The page on
734:
506:
1249:
The spherical cosine formulae were originally proved by elementary geometry and the planar cosine rule (Todhunter, Art.37). He also gives a derivation using simple coordinate geometry and the planar cosine rule (Art.60). The approach outlined here uses simpler
9327:
often small: for example the triangles of geodetic survey typically have a spherical excess much less than 1' of arc. On the Earth the excess of an equilateral triangle with sides 21.3 km (and area 393 km) is approximately 1 arc second.
9201:
9303:
345:
Sides are also expressed in radians. A side (regarded as a great circle arc) is measured by the angle that it subtends at the centre. On the unit sphere, this radian measure is numerically equal to the arc length. By convention, the sides of
8444:
10400:
8303:
4158:{\displaystyle {\begin{aligned}\cos a&=\cos b\cos c+\sin b\sin c\cos A\\&=\cos b\ (\cos a\cos b+\sin a\sin b\cos C)+\sin b\sin C\sin a\cot A\\\cos a\sin ^{2}b&=\cos b\sin a\sin b\cos C+\sin b\sin C\sin a\cot A.\end{aligned}}}
9338:
8658:
1222:
9606:
466:. This great circle is defined by the intersection of a diametral plane with the surface. Draw the normal to that plane at the centre: it intersects the surface at two points and the point that is on the same side of the plane as
6211:
9868:
6431:
First, write the six parts of the triangle (three vertex angles, three arc angles for the sides) in the order they occur around any circuit of the triangle: for the triangle shown above left, going clockwise starting with
9000:
1740:
6488:
8536:
3213:
10302:
1908:
8155:
1999:
739:
8054:{\displaystyle {\begin{aligned}\cos a&=(\cos a\,\cos c+\sin a\,\sin c\,\cos B)\cos c+\sin b\,\sin c\,\cos A\\\cos a\,\sin ^{2}c&=\sin a\,\cos c\,\sin c\,\cos B+\sin b\,\sin c\,\cos A\end{aligned}}}
5632:
5074:
2877:{\displaystyle {\begin{aligned}\cos A&=-\cos B\,\cos C+\sin B\,\sin C\,\cos a,\\\cos B&=-\cos C\,\cos A+\sin C\,\sin A\,\cos b,\\\cos C&=-\cos A\,\cos B+\sin A\,\sin B\,\cos c.\end{aligned}}}
7822:
6686:{\displaystyle {\begin{aligned}\sin a&=\tan({\tfrac {\pi }{2}}-B)\,\tan b\\&=\cos({\tfrac {\pi }{2}}-c)\,\cos({\tfrac {\pi }{2}}-A)\\&=\cot B\,\tan b\\&=\sin c\,\sin A.\end{aligned}}}
6460:
from the list. The remaining parts can then be drawn as five ordered, equal slices of a pentagram, or circle, as shown in the above figure (right). For any choice of three contiguous parts, one (the
1820:
10229:{\displaystyle \tan {\tfrac {1}{2}}E_{4}={\frac {\sin {\tfrac {1}{2}}(\varphi _{2}+\varphi _{1})}{\cos {\tfrac {1}{2}}(\varphi _{2}-\varphi _{1})}}\tan {\tfrac {1}{2}}(\lambda _{2}-\lambda _{1}).}
5054:
4982:
4288:
7240:/2 on the unit sphere the equations governing the remaining sides and angles may be obtained by applying the rules for the right spherical triangle of the previous section to the polar triangle
3855:
2640:
1397:
9129:
1610:{\displaystyle {\begin{aligned}{\vec {OA}}:&\quad (0,\,0,\,1)\\{\vec {OB}}:&\quad (\sin c,\,0,\,\cos c)\\{\vec {OC}}:&\quad (\sin b\cos A,\,\sin b\sin A,\,\cos b).\end{aligned}}}
10672:
9241:
1096:
10405:
The area of a polygon can be calculated from individual quadrangles of the above type, from (analogously) individual triangle bounded by a segment of the polygon and two meridians, by a
10034:
9988:
9557:
9601:
1982:
958:{\displaystyle {\begin{aligned}\cos a&=\cos b\cos c+\sin b\sin c\cos A,\\\cos b&=\cos c\cos a+\sin c\sin a\cos B,\\\cos c&=\cos a\cos b+\sin a\sin b\cos C.\end{aligned}}}
8315:
331:
8177:
10780:
697:{\displaystyle {\begin{alignedat}{3}A'&=\pi -a,&\qquad B'&=\pi -b,&\qquad C'&=\pi -c,\\a'&=\pi -A,&b'&=\pi -B,&c'&=\pi -C.\end{alignedat}}}
399:
337:
In particular, the sum of the angles of a spherical triangle is strictly greater than the sum of the angles of a triangle defined on the
Euclidean plane, which is always exactly
8541:
1123:
1007:
11046:
Robert G. Chamberlain, William H. Duquette, Jet
Propulsion Laboratory. The paper develops and explains many useful formulae, perhaps with a focus on navigation and cartography.
1039:
9942:
9912:
4279:
4229:
6428:
for the ten independent equations: the mnemonic is called Napier's circle or Napier's pentagon (when the circle in the above figure, right, is replaced by a pentagon).
9787:
10307:
1637:
9307:
Since the area of a triangle cannot be negative the spherical excess is always positive. It is not necessarily small, because the sum of the angles may attain 5
7287:
6701:
4989:
4176:. Similar techniques with the other two cosine rules give CT3 and CT5. The other three equations follow by applying rules 1, 3 and 5 to the polar triangle.
731:
The cosine rule is the fundamental identity of spherical trigonometry: all other identities, including the sine rule, may be derived from the cosine rule:
1825:
8071:
9496:{\displaystyle \tan {\tfrac {1}{4}}E={\sqrt {\tan {\tfrac {1}{2}}s\,\tan {\tfrac {1}{2}}(s-a)\,\tan {\tfrac {1}{2}}(s-b)\,\tan {\tfrac {1}{2}}(s-c)}}}
11018:
9748:{\displaystyle \tan {\tfrac {1}{2}}E={\frac {\tan {\frac {1}{2}}a\tan {\frac {1}{2}}b\sin C}{1+\tan {\frac {1}{2}}a\tan {\frac {1}{2}}b\cos C}}.}
11021:
by
Sudipto Banerjee. The paper derives the spherical law of cosines and law of sines using elementary linear algebra and projection matrices.
6369:{\displaystyle {\frac {\tan {\tfrac {1}{2}}(A-B)}{\tan {\tfrac {1}{2}}(A+B)}}={\frac {\tan {\tfrac {1}{2}}(a-b)}{\tan {\tfrac {1}{2}}(a+b)}}}
1745:
511:
10885:
8449:
114:. Since then, significant developments have been the application of vector methods, quaternion methods, and the use of numerical methods.
10729:
10241:
8159:
Similar substitutions in the other cosine and supplementary cosine formulae give a large variety of 5-part rules. They are rarely used.
6398:/2 the various identities given above are considerably simplified. There are ten identities relating three elements chosen from the set
110:
and others, and attained an essentially complete form by the end of the nineteenth century with the publication of
Todhunter's textbook
9102:{\displaystyle {{\text{Area of polygon}} \atop {\text{(on the unit sphere)}}}\equiv E_{N}=\left(\sum _{n=1}^{N}A_{n}\right)-(N-2)\pi .}
94:
The origins of spherical trigonometry in Greek mathematics and the major developments in
Islamic mathematics are discussed fully in
5064:
The
Delambre analogies (also called Gauss analogies) were published independently by Delambre, Gauss, and Mollweide in 1807β1809.
10948:
10847:
11015:
A free software to solve the spherical triangles, configurable to different practical applications and configured for gnomonic
10777:
10803:
8929:
Not all of the rules obtained are numerically robust in extreme examples, for example when an angle approaches zero or
289:
10760:
8933:. Problems and solutions may have to be examined carefully, particularly when writing code to solve an arbitrary triangle.
8829:
Another approach is to split the triangle into two right-angled triangles. For example, take the Case 3 example where
4995:
4923:
10868:
8804:
8669:
357:
17:
10918:
7224:
A quadrantal spherical triangle is defined to be a spherical triangle in which one of the sides subtends an angle of
10532:
9331:
There are many formulae for the excess. For example, Todhunter, (Art.101β103) gives ten examples including that of
9319:
angles). For example, an octant of a sphere is a spherical triangle with three right angles, so that the excess is
503:. A very important theorem (Todhunter, Art.27) proves that the angles and sides of the polar triangle are given by
1044:
11064:
9993:
9947:
9332:
8814:
was the first to list the six distinct cases (2-7 in the diagram) of a right triangle in spherical trigonometry.
11029:
8810:
There is a full discussion of the solution of oblique triangles in
Todhunter. See also the discussion in Ross.
9506:
242:. The sphere has a radius of 1, and so the side lengths and lower case angles are equivalent (see arc length).
6200:
1929:
971:, to which they are asymptotically equivalent in the limit of small interior angles. (On the unit sphere, if
107:
99:
9564:
194:
From this point in the article, discussion will be restricted to spherical triangles, referred to simply as
6464:
part) will be adjacent to two parts and opposite the other two parts. The ten Napier's Rules are given by
10417:
as commonly done in GIS. The other algorithms can still be used with the side lengths calculated using a
9196:{\displaystyle {{\text{Area of triangle}} \atop {\text{(on the unit sphere)}}}\equiv E=E_{3}=A+B+C-\pi ,}
974:
10842:
9887:
The spherical excess of a spherical quadrangle bounded by the equator, the two meridians of longitudes
9603:), it is often better to use the formula for the excess in terms of two edges and their included angle
5617:
Proved by expanding the numerators and using the half angle formulae. (Todhunter, Art.54 and
Delambre)
11040:, a manuscript in Arabic that dates back to 1740 and talks about spherical trigonometry, with diagrams
9298:{\displaystyle A+B+C=\pi +{\frac {4\pi \times {\text{Area of triangle}}}{\text{Area of the sphere}}}.}
3840:
To prove the first formula start from the first cosine rule and on the right-hand side substitute for
10471:
1012:
412:
before using the identities given below. Likewise, after a calculation on the unit sphere the sides
11025:
10852:
10440:
9917:
6196:
2449:
1822:
This equation can be re-arranged to give explicit expressions for the angle in terms of the sides:
1255:
1237:
1099:
726:
11049:
9890:
215:
Both vertices and angles at the vertices of a triangle are denoted by the same upper case letters
10888:
whenever the area of the triangle is small relative to the surface area of the entire Earth; see
4234:
95:
5056:
the third is a quotient and the remainder follow by applying the results to the polar triangle.
4187:
8817:
8439:{\displaystyle \cos a\cos A=-\cos B\,\cos C\,\cos a+\sin B\,\sin C-\sin B\,\sin C\,\sin ^{2}a.}
1921:
1111:
64:
10723:
10395:{\textstyle E_{4}\approx {\frac {1}{2}}(\varphi _{2}+\varphi _{1})(\lambda _{2}-\lambda _{1})}
8298:{\displaystyle \cos a\cos A=\cos b\,\cos c\,\cos A+\sin b\,\sin c-\sin b\,\sin c\,\sin ^{2}A.}
7216:
712:
for a right-angled triangle. The polar triangle of a polar triangle is the original triangle.
260:) may be regarded either as the angle between the two planes that intersect the sphere at the
122:
10692:
Todhunter, Isaac (1873). "Note on the history of certain formulæ in spherical trigonometry".
10445:
10418:
10414:
6383:
2458:
2453:
matrices and also quotes methods in differential geometry and the group theory of rotations.
404:
The sphere's radius is taken as unity. For specific practical problems on a sphere of radius
184:
10977:
10607:
10466:
10435:
8811:
8803:/2 and therefore free from ambiguity. There is a full discussion in Todhunter. The article
8653:{\displaystyle \sin b\,\sin c+\cos b\,\cos c\,\cos A=\sin B\,\sin C-\cos B\,\cos C\,\cos a}
6203:
1217:{\displaystyle {\frac {\sin A}{\sin a}}={\frac {\sin B}{\sin b}}={\frac {\sin C}{\sin c}}.}
8:
10838:
9876:
8955:
8683:
437:
206:
164:
Such polygons may have any number of sides greater than 1. Two-sided spherical polygonsβ
10997:
10664:
10656:
10583:
10461:
10410:
158:
48:
37:
10940:
10894:
10554:
7220:
A quadrantal spherical triangle together with Napier's circle for use in his mnemonics
2895:). The cotangent, or four-part, formulae relate two sides and two angles forming four
10994:
10974:
10668:
10636:
10604:
10560:
1117:
1290:
drawn from the origin to the vertices of the triangle (on the unit sphere). The arc
102:. The subject came to fruition in Early Modern times with important developments by
10701:
10652:
10648:
10589:
10568:
10456:
8967:
8946:
1251:
6192:
These identities follow by division of the
Delambre formulae. (Todhunter, Art.52)
2564:. Therefore, the invariance of the triple product under cyclic permutations gives
10872:
10797:
10784:
10754:
10522:
9863:{\displaystyle \tan {\tfrac {1}{2}}E=\tan {\tfrac {1}{2}}a\tan {\tfrac {1}{2}}b.}
1242:
52:
9219:. An earlier proof was derived, but not published, by the English mathematician
3163:
3121:
3076:
3034:
2989:
2947:
10694:
The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science
10430:
9872:
9220:
8997:-th interior angle. The area of such a polygon is given by (Todhunter, Art.99)
7228:/2 radians at the centre of the sphere: on the unit sphere the side has length
166:
10866:
10705:
8762:
and then we have Case 7 (rotated). There are either one or two solutions.
6693:
The full set of rules for the right spherical triangle is (Todhunter, Art.62)
4988:
in terms of the sides and replacing the sum of two cosines by a product. (See
11058:
11043:
10912:
10451:
10406:
9216:
1735:{\displaystyle {\vec {OB}}\cdot {\vec {OC}}=\sin c\sin b\cos A+\cos c\cos b.}
462:
is defined as follows. Consider the great circle that contains the side
126:
Eight spherical triangles defined by the intersection of three great circles.
4920:
The proof (Todhunter, Art.49) of the first formula starts from the identity
11012:
10564:
10238:
This result is obtained from one of Napier's analogies. In the limit where
9233:. The definition of the excess is independent of the radius of the sphere.
7814:
Substituting the second cosine rule into the first and simplifying gives:
6468:
sine of the middle part = the product of the tangents of the adjacent parts
3205:
and the six possible equations are (with the relevant set shown at right):
2600:
Applying the cosine rules to the polar triangle gives (Todhunter, Art.47),
2584:
1225:
968:
188:
154:
150:
146:
76:
6471:
sine of the middle part = the product of the cosines of the opposite parts
2907:). In such a set there are inner and outer parts: for example in the set (
10750:
8942:
6421:
103:
10526:
9944:
and the great-circle arc between two points with longitude and latitude
9561:
Because some triangles are badly characterized by their edges (e.g., if
8446:
Subtracting the two and noting that it follows from the sine rules that
32:
10660:
8807:
presents variants on these methods with a slightly different notation.
88:
10588:(6th ed.). Cambridge University Press. Chapter 1 – via the
11002:
10982:
10612:
8951:
8660:
which is a relation between the six parts of the spherical triangle.
8531:{\displaystyle \sin b\,\sin c\,\sin ^{2}A=\sin B\,\sin C\,\sin ^{2}a}
2427:
Since the right hand side is invariant under a cyclic permutation of
80:
72:
11037:
8726:
and then we have Case 7. There are either one or two solutions.
79:. Spherical trigonometry is of great importance for calculations in
10719:
10297:{\displaystyle \varphi _{1},\varphi _{2},\lambda _{2}-\lambda _{1}}
6425:
2583:
which is the first of the sine rules. See curved variations of the
1926:
This derivation is given in Todhunter, (Art.40). From the identity
1903:{\displaystyle \cos A={\frac {\cos a-\cos b\cos c}{\sin b\sin c}}.}
8150:{\displaystyle \cos a\sin c=\sin a\,\cos c\,\cos B+\sin b\,\cos A}
10941:"Surface area of polygon on sphere or ellipsoid β MATLAB areaint"
8706:
but, to avoid ambiguities, the half angle formulae are preferred.
268:
141:
84:
10917:. Association of American Geographers Annual Meeting. NASA JPL.
8780:
but, to avoid ambiguities, the half-side formulae are preferred.
10972:
10890:
10602:
6378:
2891:
The six parts of a triangle may be written in cyclic order as (
1228:
when the sides are much smaller than the radius of the sphere.
275:
68:
11019:"Revisiting Spherical Trigonometry with Orthogonal Projectors"
10911:
Chamberlain, Robert G.; Duquette, William H. (17 April 2007).
10637:"Revisiting Spherical Trigonometry with Orthogonal Projectors"
10304:
are all small, this reduces to the familiar trapezoidal area,
8305:
Similarly multiplying the first supplementary cosine rule by
11038:"The Book of Instruction on Deviant Planes and Simple Planes"
2506:
in the basis shown. Similarly, in a basis oriented with the
172:
56:
1912:
The other cosine rules are obtained by cyclic permutations.
10992:
8768:
The supplemental cosine rule may be used to give the sides
2927:. The cotangent rule may be written as (Todhunter, Art.44)
7211:
1742:
Equating the two expressions for the scalar product gives
112:
Spherical trigonometry for the use of colleges and Schools
9215:
of the triangle. This theorem is named after its author,
183:
One spherical polygon with interesting properties is the
11009:
a more thorough list of identities, with some derivation
10989:
a more thorough list of identities, with some derivation
10802:. Philadelphia: J. B. Lippincott & Co. p. 165.
8670:
Solution of triangles Β§ Solving spherical triangles
4917:
Another twelve identities follow by cyclic permutation.
8784:
Case 7: two angles and two opposite sides given (SSAA).
6189:
Another eight identities follow by cyclic permutation.
5614:
Another eight identities follow by cyclic permutation.
1815:{\displaystyle \cos a=\cos b\cos c+\sin b\sin c\cos A.}
271:
of the great circle arcs where they meet at the vertex.
51:
that deals with the metrical relationships between the
10310:
10183:
10133:
10084:
10050:
9843:
9822:
9798:
9617:
9567:
9517:
9465:
9431:
9397:
9375:
9349:
6598:
6567:
6517:
6337:
6302:
6260:
6225:
6159:
6123:
6088:
6049:
6023:
5987:
5952:
5913:
5888:
5852:
5817:
5778:
5752:
5716:
5681:
5642:
5583:
5548:
5518:
5483:
5451:
5416:
5386:
5351:
5320:
5285:
5255:
5220:
5187:
5152:
5122:
5087:
5014:
4942:
4795:
4683:
4581:
4493:
4400:
4298:
10774:
The Construction of the Wonderful Canon of Logarithms
10244:
10042:
9996:
9950:
9920:
9893:
9790:
9609:
9509:
9341:
9244:
9227:
both of the above area expressions are multiplied by
9207:
is the amount by which the sum of the angles exceeds
9132:
9003:
8824:
8544:
8452:
8318:
8180:
8074:
7820:
7285:
6699:
6486:
6214:
5630:
5072:
5049:{\displaystyle 2\cos ^{2}\!{\tfrac {A}{2}}=1+\cos A,}
4998:
4977:{\displaystyle 2\sin ^{2}\!{\tfrac {A}{2}}=1-\cos A,}
4926:
4286:
4237:
4190:
3853:
3211:
2933:
2638:
1997:
1932:
1828:
1748:
1640:
1395:
1126:
1047:
1015:
977:
967:
These identities generalize the cosine rule of plane
737:
509:
496:
is the polar triangle corresponding to triangle
408:
the measured lengths of the sides must be divided by
360:
292:
8756:
Case 5: two angles and an opposite side given (AAS).
8730:
Case 4: two angles and an included side given (ASA).
8720:
Case 3: two sides and an opposite angle given (SSA).
8710:
Case 2: two sides and an included angle given (SAS).
4179:
1224:
These identities approximate the sine rule of plane
10910:
10647:(5), Mathematical Association of America: 375β381,
10865:Another proof of Girard's theorem may be found at
10394:
10296:
10228:
10028:
9982:
9936:
9906:
9862:
9747:
9595:
9551:
9495:
9297:
9195:
9101:
8652:
8530:
8438:
8297:
8149:
8053:
7799:
7201:
6685:
6440:. Next replace the parts that are not adjacent to
6368:
6179:
5606:
5048:
4976:
4907:
4273:
4223:
4157:
3832:
3197:
2876:
2419:
1976:
1902:
1814:
1734:
1609:
1216:
1090:
1033:
1001:
957:
696:
393:
325:
282:spherical triangles are (by convention) less than
10836:
9111:For the case of a spherical triangle with angles
8958:: the triangles of constant area on a fixed base
8805:Solution of triangles#Solving spherical triangles
6456:) by their complements and then delete the angle
5012:
4940:
3187:
3156:
3153:
3145:
3114:
3111:
3100:
3069:
3066:
3058:
3027:
3024:
3013:
2982:
2979:
2971:
2940:
2937:
2341:
2324:
2307:
2244:
2229:
11056:
8883:. Then use Napier's rules to solve the triangle
1387:. Therefore, the three vectors have components:
1231:
40:is a spherical triangle with three right angles.
8694:The cosine rule may be used to give the angles
4992:.) The second formula starts from the identity
2886:
10799:A Treatise on Plane and Spherical Trigonometry
9882:
1915:
1254:methods. (These methods are also discussed at
715:
27:Geometry of figures on the surface of a sphere
2439:the spherical sine rule follows immediately.
267:, or, equivalently, as the angle between the
10517:
10515:
10513:
10511:
10509:
10507:
8936:
6379:Napier's rules for right spherical triangles
2595:
1091:{\displaystyle (\cos a-\cos b)^{2}\approx 0}
145:on the surface of the sphere. Its sides are
10505:
10503:
10501:
10499:
10497:
10495:
10493:
10491:
10489:
10487:
10029:{\displaystyle (\lambda _{2},\varphi _{2})}
9983:{\displaystyle (\lambda _{1},\varphi _{1})}
8924:
8853:. Use Napier's rules to solve the triangle
8841:are given. Construct the great circle from
8732:The four-part cotangent formulae for sets (
10756:Mirifici Logarithmorum Canonis Constructio
2442:
11050:Online computation of spherical triangles
10886:Legendre's theorem on spherical triangles
10795:
10691:
10521:
9457:
9423:
9389:
8640:
8630:
8608:
8586:
8576:
8554:
8511:
8501:
8472:
8462:
8416:
8406:
8384:
8362:
8352:
8275:
8265:
8243:
8221:
8211:
8137:
8115:
8105:
8037:
8027:
8005:
7995:
7985:
7952:
7929:
7919:
7885:
7875:
7853:
7780:
7729:
7679:
7631:
7581:
7533:
7483:
7432:
7382:
7327:
7182:
7134:
7084:
7036:
6986:
6938:
6888:
6840:
6790:
6738:
6666:
6637:
6587:
6537:
3793:
3771:
3745:
3691:
3669:
3643:
3589:
3567:
3541:
3487:
3465:
3439:
3385:
3363:
3337:
3282:
3260:
3234:
2857:
2847:
2825:
2780:
2770:
2748:
2703:
2693:
2671:
2233:
1587:
1565:
1499:
1492:
1441:
1434:
704:Therefore, if any identity is proved for
230:Sides are denoted by lower-case letters:
11044:Some Algorithms for Polygons on a Sphere
10914:Some algorithms for polygons on a sphere
10718:
10634:
10484:
9764:is a right triangle with right angle at
9552:{\displaystyle s={\tfrac {1}{2}}(a+b+c)}
8950:
8816:
8682:
8663:
7215:
6382:
2899:parts around the triangle, for example (
1241:
436:
205:
121:
31:
10848:MacTutor History of Mathematics Archive
10628:
9596:{\textstyle a=b\approx {\frac {1}{2}}c}
8794:; or, use Case 3 (SSA) or case 5 (AAS).
7212:Napier's rules for quadrantal triangles
1977:{\displaystyle \sin ^{2}A=1-\cos ^{2}A}
470:is (conventionally) termed the pole of
14:
11057:
10889:
10878:
10749:
10552:
9236:The converse result may be written as
8167:Multiplying the first cosine rule by
6394:, of a spherical triangle is equal to
326:{\displaystyle \pi <A+B+C<3\pi }
153:βthe spherical geometry equivalent of
10993:
10973:
10603:
10581:
8162:
6195:Taking quotients of these yields the
5620:
5059:
130:
11026:"A Visual Proof of Girard's Theorem"
8716:and then we are back to Case 1.
8674:
210:The basic triangle on a unit sphere.
191:with a right angle at every vertex.
2587:to see details of this derivation.
1324:. Introduce a Cartesian basis with
394:{\displaystyle 0<a+b+c<2\pi }
24:
9134:
9005:
8825:Solution by right-angled triangles
7809:
4165:The result follows on dividing by
1002:{\displaystyle a,b,c\rightarrow 0}
432:
350:spherical triangles are less than
25:
11076:
10966:
10921:from the original on 22 July 2020
9323:/2. In practical applications it
8986:-sided spherical polygon and let
8766:Case 6: three angles given (AAA).
4984:using the cosine rule to express
4180:Half-angle and half-side formulae
3162:
3120:
3075:
3033:
2988:
2946:
10585:Text-Book on Spherical Astronomy
8970:through the points antipodal to
8692:Case 1: three sides given (SSS).
1984:and the explicit expression for
117:
63:, traditionally expressed using
10951:from the original on 2021-05-01
10933:
10904:
10859:
10830:
10817:
10806:from the original on 2021-07-11
10776:is available as en e-book from
10763:from the original on 2013-04-30
10732:from the original on 2020-07-22
10675:from the original on 2020-07-22
10641:The College Mathematics Journal
10553:Clarke, Alexander Ross (1880).
10535:from the original on 2020-04-14
7353:
7343:
7342:
7298:
7232:/2. In the case that the side
6764:
6754:
6753:
6712:
6039:
5768:
5470:
5207:
5206:
4392:
4391:
3292:
3224:
1540:
1476:
1424:
1296:subtends an angle of magnitude
1034:{\displaystyle \sin a\approx a}
720:
573:
543:
11030:Wolfram Demonstrations Project
10789:
10743:
10712:
10685:
10653:10.1080/07468342.2004.11922099
10596:
10575:
10546:
10389:
10363:
10360:
10334:
10220:
10194:
10170:
10144:
10121:
10095:
10023:
9997:
9977:
9951:
9546:
9528:
9488:
9476:
9454:
9442:
9420:
9408:
9090:
9078:
8799:half-angles will be less than
7895:
7841:
6615:
6594:
6584:
6563:
6534:
6513:
6360:
6348:
6325:
6313:
6283:
6271:
6248:
6236:
6146:
6134:
6111:
6099:
6072:
6060:
6010:
5998:
5975:
5963:
5936:
5924:
5875:
5863:
5840:
5828:
5801:
5789:
5739:
5727:
5704:
5692:
5665:
5653:
5571:
5559:
5506:
5494:
5439:
5427:
5374:
5362:
5308:
5296:
5243:
5231:
5175:
5163:
5110:
5098:
4894:
4882:
4873:
4861:
4850:
4838:
4779:
4767:
4747:
4735:
4726:
4714:
4645:
4633:
4624:
4612:
4545:
4533:
4455:
4443:
4362:
4350:
4341:
4329:
4265:
4247:
4218:
4200:
3995:
3944:
3823:
3808:
3721:
3706:
3619:
3604:
3517:
3502:
3415:
3400:
3313:
3298:
2208:
2174:
2168:
2143:
2140:
2115:
1672:
1652:
1634:in terms of the components is
1597:
1541:
1527:
1509:
1477:
1463:
1445:
1425:
1411:
1073:
1048:
993:
13:
1:
10477:
9937:{\displaystyle \lambda _{2},}
2590:
1375:-plane and the angle between
1232:Derivation of the cosine rule
1105:
100:Mathematics in medieval Islam
9907:{\displaystyle \lambda _{1}}
8538:produces Cagnoli's equation
6390:When one of the angles, say
3847:from the third cosine rule:
2887:Cotangent four-part formulae
1300:at the centre and therefore
1261:Consider three unit vectors
7:
10796:Chauvenet, William (1867).
10559:. Oxford: Clarendon Press.
10531:(5th ed.). MacMillan.
10424:
9883:From latitude and longitude
8845:that is normal to the side
8786:Use Napier's analogies for
8752:follows from the sine rule.
4274:{\displaystyle 2S=(A+B+C),}
1916:Derivation of the sine rule
716:Cosine rules and sine rules
455:associated with a triangle
424:must be multiplied by
201:
10:
11081:
10725:Connaissance des Tems 1809
10635:Banerjee, Sudipto (2004),
8940:
8667:
4224:{\displaystyle 2s=(a+b+c)}
1919:
1235:
1109:
724:
10825:Master Math: Trigonometry
10706:10.1080/14786447308640820
10472:Triangulation (surveying)
9875:is defined similarly for
8937:Area and spherical excess
8061:Cancelling the factor of
4990:sum-to-product identities
2596:Supplemental cosine rules
10978:"Spherical Trigonometry"
10853:University of St Andrews
10608:"Spherical Trigonometry"
10441:Ellipsoidal trigonometry
9223:. On a sphere of radius
8925:Numerical considerations
2450:Spherical law of cosines
1991:given immediately above
1256:Spherical law of cosines
1238:Spherical law of cosines
1120:is given by the formula
1100:Spherical law of cosines
727:Spherical law of cosines
274:Angles are expressed in
8962:have their free vertex
2443:Alternative derivations
1350:-plane making an angle
486:are defined similarly.
401:(Todhunter, Art.22,32).
333:(Todhunter, Art.22,32).
96:History of trigonometry
65:trigonometric functions
11065:Spherical trigonometry
10891:Clarke, Alexander Ross
10843:"Nasir al-Din al-Tusi"
10528:Spherical Trigonometry
10396:
10298:
10230:
10030:
9984:
9938:
9908:
9864:
9749:
9597:
9553:
9497:
9299:
9197:
9103:
9059:
8979:
8821:
8712:The cosine rule gives
8687:
8654:
8532:
8440:
8299:
8151:
8055:
7801:
7279:etc. The results are:
7221:
7203:
6687:
6387:
6370:
6181:
5608:
5050:
4978:
4909:
4275:
4225:
4159:
3834:
3199:
2878:
2421:
1978:
1922:Spherical law of sines
1904:
1816:
1736:
1611:
1246:
1218:
1112:Spherical law of sines
1092:
1035:
1003:
959:
698:
448:
395:
327:
211:
187:, a 5-sided spherical
127:
45:Spherical trigonometry
41:
10827:, Career Press, 2002.
10448:or spherical distance
10446:Great-circle distance
10419:great-circle distance
10415:equal-area projection
10397:
10299:
10231:
10031:
9985:
9939:
9909:
9865:
9784:, so this reduces to
9750:
9598:
9554:
9498:
9300:
9198:
9104:
9039:
8954:
8820:
8686:
8664:Solution of triangles
8655:
8533:
8441:
8300:
8152:
8056:
7802:
7219:
7204:
6688:
6386:
6371:
6201:Persian mathematician
6182:
5609:
5051:
4979:
4910:
4276:
4226:
4160:
3835:
3200:
2919:, the outer angle is
2911:) the inner angle is
2879:
2519:, the triple product
2459:scalar triple product
2422:
1979:
1905:
1817:
1737:
1612:
1245:
1219:
1093:
1036:
1004:
960:
699:
474:and it is denoted by
440:
396:
328:
209:
185:pentagramma mirificum
125:
35:
10998:"Spherical Triangle"
10839:Robertson, Edmund F.
10772:An 1889 translation
10582:Smart, W.M. (1977).
10467:Spherical polyhedron
10436:Celestial navigation
10308:
10242:
10040:
9994:
9948:
9918:
9891:
9788:
9607:
9565:
9507:
9339:
9242:
9211:radians, called the
9142:(on the unit sphere)
9130:
9013:(on the unit sphere)
9001:
8921:follow by addition.
8812:Nasir al-Din al-Tusi
8758:The sine rule gives
8722:The sine rule gives
8542:
8450:
8316:
8178:
8072:
7818:
7283:
6697:
6484:
6424:provided an elegant
6212:
6204:Nasir al-Din al-Tusi
5628:
5070:
4996:
4924:
4284:
4235:
4188:
3851:
3209:
2931:
2923:, the outer side is
2915:, the inner side is
2636:
1995:
1930:
1826:
1746:
1638:
1393:
1124:
1045:
1013:
975:
735:
507:
358:
290:
10901:(Chapters 2 and 9).
10837:O'Connor, John J.;
9877:hyperbolic geometry
1619:The scalar product
441:The polar triangle
61:spherical triangles
18:Spherical triangles
10995:Weisstein, Eric W.
10975:Weisstein, Eric W.
10899:. Clarendon Press.
10884:This follows from
10871:2012-10-31 at the
10823:Ross, Debra Anne.
10783:2020-03-03 at the
10720:Delambre, J. B. J.
10605:Weisstein, Eric W.
10462:Spherical geometry
10392:
10294:
10226:
10192:
10142:
10093:
10059:
10026:
9980:
9934:
9904:
9860:
9852:
9831:
9807:
9745:
9626:
9593:
9549:
9526:
9493:
9474:
9440:
9406:
9384:
9358:
9295:
9288:Area of the sphere
9193:
9099:
8980:
8868:to find the sides
8822:
8688:
8650:
8528:
8436:
8295:
8163:Cagnoli's Equation
8147:
8051:
8049:
7797:
7795:
7222:
7199:
7197:
6683:
6681:
6607:
6576:
6526:
6388:
6366:
6346:
6311:
6269:
6234:
6199:, first stated by
6177:
6175:
6168:
6132:
6097:
6058:
6032:
5996:
5961:
5922:
5897:
5861:
5826:
5787:
5761:
5725:
5690:
5651:
5621:Napier's analogies
5604:
5602:
5592:
5557:
5527:
5492:
5460:
5425:
5395:
5360:
5329:
5294:
5264:
5229:
5196:
5161:
5131:
5096:
5060:Delambre analogies
5046:
5023:
4974:
4951:
4905:
4903:
4804:
4692:
4590:
4502:
4409:
4307:
4271:
4221:
4155:
4153:
3830:
3828:
3195:
3183:
3182:
3141:
3140:
3096:
3095:
3054:
3053:
3009:
3008:
2967:
2966:
2874:
2872:
2417:
2415:
1974:
1900:
1812:
1732:
1607:
1605:
1358:-axis. The vector
1247:
1214:
1088:
1031:
999:
955:
953:
694:
692:
449:
391:
323:
212:
131:Spherical polygons
128:
49:spherical geometry
42:
38:octant of a sphere
10945:www.mathworks.com
10332:
10191:
10174:
10141:
10092:
10058:
9851:
9830:
9806:
9740:
9725:
9706:
9670:
9651:
9625:
9588:
9525:
9491:
9473:
9439:
9405:
9383:
9357:
9290:
9289:
9284:
9145:
9143:
9138:
9016:
9014:
9009:
8898:to find the side
8675:Oblique triangles
7748:
7700:
7650:
7602:
7552:
7504:
7451:
7403:
7349:
7294:
7153:
7105:
7055:
7007:
6957:
6909:
6859:
6811:
6760:
6708:
6606:
6575:
6525:
6364:
6345:
6310:
6287:
6268:
6233:
6167:
6150:
6131:
6096:
6057:
6031:
6014:
5995:
5960:
5921:
5896:
5879:
5860:
5825:
5786:
5760:
5743:
5724:
5689:
5650:
5598:
5591:
5556:
5533:
5526:
5491:
5466:
5459:
5424:
5401:
5394:
5359:
5335:
5328:
5293:
5270:
5263:
5228:
5202:
5195:
5160:
5137:
5130:
5095:
5022:
4950:
4899:
4898:
4803:
4784:
4783:
4691:
4670:
4669:
4589:
4570:
4569:
4501:
4480:
4479:
4408:
4387:
4386:
4306:
3943:
3731:
3629:
3527:
3425:
3323:
3219:
3178:
3169:
3136:
3127:
3091:
3082:
3049:
3040:
3004:
2995:
2962:
2953:
2544:
2535:
2526:
2516:
2486:
2477:
2468:
2408:
2378:
2279:
2249:
2090:
1895:
1675:
1655:
1631:
1625:
1530:
1466:
1414:
1364:
1343:
1330:
1316:
1308:
1286:
1277:
1268:
1209:
1180:
1151:
137:spherical polygon
47:is the branch of
16:(Redirected from
11072:
11033:
11008:
11007:
10988:
10987:
10960:
10959:
10957:
10956:
10937:
10931:
10930:
10928:
10926:
10908:
10902:
10900:
10882:
10876:
10863:
10857:
10855:
10834:
10828:
10821:
10815:
10814:
10812:
10811:
10793:
10787:
10771:
10769:
10768:
10747:
10741:
10740:
10738:
10737:
10716:
10710:
10709:
10689:
10683:
10682:
10681:
10680:
10632:
10626:
10625:
10624:
10622:
10620:
10600:
10594:
10593:
10590:Internet Archive
10579:
10573:
10572:
10569:Internet Archive
10567:– via the
10550:
10544:
10543:
10541:
10540:
10519:
10457:Schwarz triangle
10401:
10399:
10398:
10393:
10388:
10387:
10375:
10374:
10359:
10358:
10346:
10345:
10333:
10325:
10320:
10319:
10303:
10301:
10300:
10295:
10293:
10292:
10280:
10279:
10267:
10266:
10254:
10253:
10235:
10233:
10232:
10227:
10219:
10218:
10206:
10205:
10193:
10184:
10175:
10173:
10169:
10168:
10156:
10155:
10143:
10134:
10124:
10120:
10119:
10107:
10106:
10094:
10085:
10075:
10070:
10069:
10060:
10051:
10035:
10033:
10032:
10027:
10022:
10021:
10009:
10008:
9989:
9987:
9986:
9981:
9976:
9975:
9963:
9962:
9943:
9941:
9940:
9935:
9930:
9929:
9913:
9911:
9910:
9905:
9903:
9902:
9869:
9867:
9866:
9861:
9853:
9844:
9832:
9823:
9808:
9799:
9783:
9775:
9767:
9763:
9754:
9752:
9751:
9746:
9741:
9739:
9726:
9718:
9707:
9699:
9684:
9671:
9663:
9652:
9644:
9635:
9627:
9618:
9602:
9600:
9599:
9594:
9589:
9581:
9558:
9556:
9555:
9550:
9527:
9518:
9502:
9500:
9499:
9494:
9492:
9475:
9466:
9441:
9432:
9407:
9398:
9385:
9376:
9367:
9359:
9350:
9322:
9314:
9310:
9304:
9302:
9301:
9296:
9291:
9287:
9286:
9285:
9283:Area of triangle
9282:
9270:
9232:
9226:
9213:spherical excess
9210:
9206:
9202:
9200:
9199:
9194:
9165:
9164:
9146:
9144:
9141:
9139:
9137:Area of triangle
9136:
9125:Girard's theorem
9123:this reduces to
9122:
9118:
9114:
9108:
9106:
9105:
9100:
9074:
9070:
9069:
9068:
9058:
9053:
9030:
9029:
9017:
9015:
9012:
9010:
9007:
8996:
8992:
8985:
8977:
8973:
8965:
8961:
8956:Lexell's theorem
8947:Geodesic polygon
8932:
8920:
8916:
8912:
8905:
8901:
8897:
8893:
8889:
8882:
8875:
8871:
8867:
8863:
8859:
8852:
8848:
8844:
8840:
8836:
8832:
8802:
8793:
8789:
8779:
8775:
8771:
8761:
8751:
8747:
8743:
8739:
8735:
8725:
8715:
8705:
8701:
8697:
8659:
8657:
8656:
8651:
8537:
8535:
8534:
8529:
8521:
8520:
8482:
8481:
8445:
8443:
8442:
8437:
8426:
8425:
8311:
8304:
8302:
8301:
8296:
8285:
8284:
8173:
8156:
8154:
8153:
8148:
8067:
8060:
8058:
8057:
8052:
8050:
7962:
7961:
7806:
7804:
7803:
7798:
7796:
7749:
7746:
7743:
7701:
7698:
7695:
7651:
7648:
7645:
7603:
7600:
7597:
7553:
7550:
7547:
7505:
7502:
7499:
7452:
7449:
7446:
7404:
7401:
7398:
7350:
7347:
7295:
7292:
7289:
7278:
7273:
7264:
7259:
7250:
7246:
7239:
7235:
7231:
7227:
7208:
7206:
7205:
7200:
7198:
7154:
7151:
7148:
7106:
7103:
7100:
7056:
7053:
7050:
7008:
7005:
7002:
6958:
6955:
6952:
6910:
6907:
6904:
6860:
6857:
6854:
6812:
6809:
6806:
6761:
6758:
6709:
6706:
6703:
6692:
6690:
6689:
6684:
6682:
6650:
6621:
6608:
6599:
6577:
6568:
6550:
6527:
6518:
6479:
6459:
6455:
6451:
6447:
6443:
6439:
6435:
6417:
6413:
6409:
6405:
6401:
6397:
6393:
6375:
6373:
6372:
6367:
6365:
6363:
6347:
6338:
6328:
6312:
6303:
6293:
6288:
6286:
6270:
6261:
6251:
6235:
6226:
6216:
6186:
6184:
6183:
6178:
6176:
6169:
6160:
6151:
6149:
6133:
6124:
6114:
6098:
6089:
6079:
6059:
6050:
6033:
6024:
6015:
6013:
5997:
5988:
5978:
5962:
5953:
5943:
5923:
5914:
5898:
5889:
5880:
5878:
5862:
5853:
5843:
5827:
5818:
5808:
5788:
5779:
5762:
5753:
5744:
5742:
5726:
5717:
5707:
5691:
5682:
5672:
5652:
5643:
5613:
5611:
5610:
5605:
5603:
5599:
5597:
5593:
5584:
5574:
5558:
5549:
5539:
5534:
5532:
5528:
5519:
5509:
5493:
5484:
5474:
5467:
5465:
5461:
5452:
5442:
5426:
5417:
5407:
5402:
5400:
5396:
5387:
5377:
5361:
5352:
5342:
5336:
5334:
5330:
5321:
5311:
5295:
5286:
5276:
5271:
5269:
5265:
5256:
5246:
5230:
5221:
5211:
5203:
5201:
5197:
5188:
5178:
5162:
5153:
5143:
5138:
5136:
5132:
5123:
5113:
5097:
5088:
5078:
5055:
5053:
5052:
5047:
5024:
5015:
5011:
5010:
4987:
4983:
4981:
4980:
4975:
4952:
4943:
4939:
4938:
4914:
4912:
4911:
4906:
4904:
4900:
4897:
4853:
4818:
4817:
4805:
4796:
4785:
4782:
4750:
4706:
4705:
4693:
4684:
4671:
4668:
4648:
4604:
4603:
4591:
4582:
4571:
4568:
4548:
4516:
4515:
4503:
4494:
4481:
4478:
4458:
4423:
4422:
4410:
4401:
4388:
4385:
4365:
4321:
4320:
4308:
4299:
4280:
4278:
4277:
4272:
4230:
4228:
4227:
4222:
4175:
4164:
4162:
4161:
4156:
4154:
4059:
4058:
3941:
3925:
3846:
3839:
3837:
3836:
3831:
3829:
3804:
3734:
3732:
3729:
3702:
3632:
3630:
3627:
3600:
3530:
3528:
3525:
3498:
3428:
3426:
3423:
3396:
3326:
3324:
3321:
3294:
3222:
3220:
3217:
3204:
3202:
3201:
3196:
3191:
3190:
3184:
3179:
3176:
3170:
3167:
3160:
3159:
3149:
3148:
3142:
3137:
3134:
3128:
3125:
3118:
3117:
3104:
3103:
3097:
3092:
3089:
3083:
3080:
3073:
3072:
3062:
3061:
3055:
3050:
3047:
3041:
3038:
3031:
3030:
3017:
3016:
3010:
3005:
3002:
2996:
2993:
2986:
2985:
2975:
2974:
2968:
2963:
2960:
2954:
2951:
2944:
2943:
2926:
2922:
2918:
2914:
2910:
2906:
2902:
2894:
2883:
2881:
2880:
2875:
2873:
2631:
2626:
2621:
2617:
2612:
2607:
2582:
2563:
2548:
2545:
2542:
2536:
2533:
2527:
2524:
2518:
2517:
2514:
2509:
2505:
2490:
2487:
2484:
2478:
2475:
2469:
2466:
2438:
2434:
2430:
2426:
2424:
2423:
2418:
2416:
2409:
2407:
2340:
2339:
2323:
2322:
2306:
2305:
2290:
2289:
2280:
2278:
2267:
2256:
2250:
2248:
2243:
2242:
2228:
2227:
2217:
2216:
2215:
2161:
2160:
2133:
2132:
2113:
2105:
2101:
2100:
2095:
2091:
2089:
2069:
2037:
2011:
2010:
1990:
1983:
1981:
1980:
1975:
1967:
1966:
1942:
1941:
1909:
1907:
1906:
1901:
1896:
1894:
1874:
1842:
1821:
1819:
1818:
1813:
1741:
1739:
1738:
1733:
1677:
1676:
1671:
1663:
1657:
1656:
1651:
1643:
1633:
1632:
1629:
1626:
1623:
1616:
1614:
1613:
1608:
1606:
1532:
1531:
1526:
1518:
1468:
1467:
1462:
1454:
1416:
1415:
1410:
1402:
1386:
1382:
1378:
1374:
1370:
1366:
1365:
1362:
1357:
1353:
1349:
1345:
1344:
1341:
1336:
1332:
1331:
1328:
1323:
1317:
1314:
1309:
1306:
1299:
1295:
1294:
1289:
1287:
1284:
1278:
1275:
1269:
1266:
1223:
1221:
1220:
1215:
1210:
1208:
1197:
1186:
1181:
1179:
1168:
1157:
1152:
1150:
1139:
1128:
1097:
1095:
1094:
1089:
1081:
1080:
1040:
1038:
1037:
1032:
1008:
1006:
1005:
1000:
964:
962:
961:
956:
954:
710:
703:
701:
700:
695:
693:
670:
641:
612:
581:
551:
521:
502:
495:
485:
481:
477:
473:
469:
465:
461:
447:
427:
423:
419:
415:
411:
407:
400:
398:
397:
392:
353:
340:
332:
330:
329:
324:
285:
278:. The angles of
266:
259:
255:
251:
241:
237:
233:
226:
222:
218:
21:
11080:
11079:
11075:
11074:
11073:
11071:
11070:
11069:
11055:
11054:
11024:
10969:
10964:
10963:
10954:
10952:
10939:
10938:
10934:
10924:
10922:
10909:
10905:
10883:
10879:
10873:Wayback Machine
10864:
10860:
10835:
10831:
10822:
10818:
10809:
10807:
10794:
10790:
10785:Wayback Machine
10766:
10764:
10748:
10744:
10735:
10733:
10728:. p. 445.
10717:
10713:
10700:(298): 98β100.
10690:
10686:
10678:
10676:
10633:
10629:
10618:
10616:
10601:
10597:
10580:
10576:
10551:
10547:
10538:
10536:
10520:
10485:
10480:
10427:
10411:Green's theorem
10383:
10379:
10370:
10366:
10354:
10350:
10341:
10337:
10324:
10315:
10311:
10309:
10306:
10305:
10288:
10284:
10275:
10271:
10262:
10258:
10249:
10245:
10243:
10240:
10239:
10214:
10210:
10201:
10197:
10182:
10164:
10160:
10151:
10147:
10132:
10125:
10115:
10111:
10102:
10098:
10083:
10076:
10074:
10065:
10061:
10049:
10041:
10038:
10037:
10017:
10013:
10004:
10000:
9995:
9992:
9991:
9971:
9967:
9958:
9954:
9949:
9946:
9945:
9925:
9921:
9919:
9916:
9915:
9898:
9894:
9892:
9889:
9888:
9885:
9842:
9821:
9797:
9789:
9786:
9785:
9777:
9769:
9765:
9758:
9717:
9698:
9685:
9662:
9643:
9636:
9634:
9616:
9608:
9605:
9604:
9580:
9566:
9563:
9562:
9516:
9508:
9505:
9504:
9464:
9430:
9396:
9374:
9366:
9348:
9340:
9337:
9336:
9320:
9312:
9308:
9281:
9271:
9269:
9243:
9240:
9239:
9228:
9224:
9208:
9204:
9160:
9156:
9140:
9135:
9133:
9131:
9128:
9127:
9120:
9116:
9112:
9064:
9060:
9054:
9043:
9038:
9034:
9025:
9021:
9011:
9008:Area of polygon
9006:
9004:
9002:
8999:
8998:
8994:
8991:
8987:
8983:
8975:
8971:
8963:
8959:
8949:
8939:
8930:
8927:
8918:
8914:
8907:
8903:
8902:and the angles
8899:
8895:
8891:
8884:
8877:
8873:
8869:
8865:
8861:
8854:
8850:
8846:
8842:
8838:
8834:
8830:
8827:
8800:
8791:
8787:
8777:
8773:
8769:
8759:
8749:
8745:
8741:
8737:
8733:
8723:
8713:
8703:
8699:
8695:
8677:
8672:
8666:
8543:
8540:
8539:
8516:
8512:
8477:
8473:
8451:
8448:
8447:
8421:
8417:
8317:
8314:
8313:
8306:
8280:
8276:
8179:
8176:
8175:
8168:
8165:
8073:
8070:
8069:
8062:
8048:
8047:
7969:
7957:
7953:
7940:
7939:
7834:
7821:
7819:
7816:
7815:
7812:
7810:Five-part rules
7794:
7793:
7761:
7750:
7745:
7742:
7713:
7702:
7697:
7693:
7692:
7663:
7652:
7647:
7644:
7615:
7604:
7599:
7595:
7594:
7565:
7554:
7549:
7546:
7517:
7506:
7501:
7497:
7496:
7464:
7453:
7448:
7445:
7416:
7405:
7400:
7396:
7395:
7363:
7351:
7346:
7344:
7340:
7308:
7296:
7291:
7286:
7284:
7281:
7280:
7271:
7266:
7257:
7252:
7248:
7241:
7237:
7233:
7229:
7225:
7214:
7196:
7195:
7166:
7155:
7150:
7147:
7118:
7107:
7102:
7098:
7097:
7068:
7057:
7052:
7049:
7020:
7009:
7004:
7000:
6999:
6970:
6959:
6954:
6951:
6922:
6911:
6906:
6902:
6901:
6872:
6861:
6856:
6853:
6824:
6813:
6808:
6804:
6803:
6774:
6762:
6757:
6755:
6751:
6722:
6710:
6705:
6700:
6698:
6695:
6694:
6680:
6679:
6648:
6647:
6619:
6618:
6597:
6566:
6548:
6547:
6516:
6500:
6487:
6485:
6482:
6481:
6477:
6457:
6453:
6449:
6445:
6441:
6437:
6433:
6415:
6411:
6407:
6403:
6399:
6395:
6391:
6381:
6336:
6329:
6301:
6294:
6292:
6259:
6252:
6224:
6217:
6215:
6213:
6210:
6209:
6197:law of tangents
6174:
6173:
6158:
6122:
6115:
6087:
6080:
6078:
6048:
6040:
6037:
6022:
5986:
5979:
5951:
5944:
5942:
5912:
5903:
5902:
5887:
5851:
5844:
5816:
5809:
5807:
5777:
5769:
5766:
5751:
5715:
5708:
5680:
5673:
5671:
5641:
5631:
5629:
5626:
5625:
5623:
5601:
5600:
5582:
5575:
5547:
5540:
5538:
5517:
5510:
5482:
5475:
5473:
5471:
5468:
5450:
5443:
5415:
5408:
5406:
5385:
5378:
5350:
5343:
5341:
5338:
5337:
5319:
5312:
5284:
5277:
5275:
5254:
5247:
5219:
5212:
5210:
5208:
5204:
5186:
5179:
5151:
5144:
5142:
5121:
5114:
5086:
5079:
5077:
5073:
5071:
5068:
5067:
5062:
5013:
5006:
5002:
4997:
4994:
4993:
4985:
4941:
4934:
4930:
4925:
4922:
4921:
4902:
4901:
4854:
4819:
4816:
4809:
4794:
4786:
4751:
4707:
4704:
4697:
4682:
4673:
4672:
4649:
4605:
4602:
4595:
4580:
4572:
4549:
4517:
4514:
4507:
4492:
4483:
4482:
4459:
4424:
4421:
4414:
4399:
4389:
4366:
4322:
4319:
4312:
4297:
4287:
4285:
4282:
4281:
4236:
4233:
4232:
4189:
4186:
4185:
4182:
4166:
4152:
4151:
4066:
4054:
4050:
4038:
4037:
3923:
3922:
3867:
3854:
3852:
3849:
3848:
3841:
3827:
3826:
3803:
3755:
3733:
3728:
3725:
3724:
3701:
3653:
3631:
3626:
3623:
3622:
3599:
3551:
3529:
3524:
3521:
3520:
3497:
3449:
3427:
3422:
3419:
3418:
3395:
3347:
3325:
3320:
3317:
3316:
3293:
3244:
3221:
3216:
3212:
3210:
3207:
3206:
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3175:
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3171:
3166:
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3119:
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3112:
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3084:
3079:
3074:
3068:
3067:
3057:
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3032:
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3025:
3012:
3011:
3007:
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2998:
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2965:
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2624:
2623:
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2610:
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2605:
2598:
2593:
2565:
2550:
2549:, evaluates to
2541:
2532:
2523:
2520:
2513:
2511:
2507:
2492:
2483:
2474:
2465:
2462:
2445:
2436:
2432:
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2379:
2335:
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2318:
2314:
2301:
2297:
2288:
2281:
2268:
2257:
2255:
2252:
2251:
2238:
2234:
2223:
2219:
2218:
2211:
2207:
2156:
2152:
2128:
2124:
2114:
2112:
2103:
2102:
2096:
2070:
2038:
2036:
2032:
2031:
2018:
2006:
2002:
1998:
1996:
1993:
1992:
1985:
1962:
1958:
1937:
1933:
1931:
1928:
1927:
1924:
1918:
1875:
1843:
1841:
1827:
1824:
1823:
1747:
1744:
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1664:
1662:
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1644:
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1628:
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1536:
1519:
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1513:
1512:
1472:
1455:
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1420:
1403:
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1400:
1396:
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1390:
1384:
1380:
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1368:
1361:
1359:
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1301:
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1291:
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1240:
1234:
1198:
1187:
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1140:
1129:
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1125:
1122:
1121:
1114:
1108:
1076:
1072:
1046:
1043:
1042:
1014:
1011:
1010:
976:
973:
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952:
951:
893:
881:
880:
822:
810:
809:
751:
738:
736:
733:
732:
729:
723:
718:
705:
691:
690:
671:
663:
661:
642:
634:
632:
613:
605:
602:
601:
582:
574:
571:
552:
544:
541:
522:
514:
510:
508:
505:
504:
497:
490:
483:
479:
475:
471:
467:
463:
456:
442:
435:
433:Polar triangles
425:
421:
417:
413:
409:
405:
359:
356:
355:
351:
338:
291:
288:
287:
283:
264:
257:
253:
252:(respectively,
249:
239:
235:
231:
224:
220:
216:
204:
133:
120:
28:
23:
22:
15:
12:
11:
5:
11078:
11068:
11067:
11053:
11052:
11047:
11041:
11035:
11022:
11016:
11010:
10990:
10968:
10967:External links
10965:
10962:
10961:
10932:
10903:
10877:
10858:
10829:
10816:
10788:
10759:. p. 50.
10742:
10711:
10684:
10627:
10595:
10574:
10545:
10482:
10481:
10479:
10476:
10475:
10474:
10469:
10464:
10459:
10454:
10449:
10443:
10438:
10433:
10431:Air navigation
10426:
10423:
10391:
10386:
10382:
10378:
10373:
10369:
10365:
10362:
10357:
10353:
10349:
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10340:
10336:
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10323:
10318:
10314:
10291:
10287:
10283:
10278:
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10270:
10265:
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10225:
10222:
10217:
10213:
10209:
10204:
10200:
10196:
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10178:
10172:
10167:
10163:
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10146:
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10128:
10123:
10118:
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10073:
10068:
10064:
10057:
10054:
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10045:
10025:
10020:
10016:
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10007:
10003:
9999:
9979:
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9970:
9966:
9961:
9957:
9953:
9933:
9928:
9924:
9901:
9897:
9884:
9881:
9859:
9856:
9850:
9847:
9841:
9838:
9835:
9829:
9826:
9820:
9817:
9814:
9811:
9805:
9802:
9796:
9793:
9757:When triangle
9744:
9738:
9735:
9732:
9729:
9724:
9721:
9716:
9713:
9710:
9705:
9702:
9697:
9694:
9691:
9688:
9683:
9680:
9677:
9674:
9669:
9666:
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9639:
9633:
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9624:
9621:
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9612:
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9490:
9487:
9484:
9481:
9478:
9472:
9469:
9463:
9460:
9456:
9453:
9450:
9447:
9444:
9438:
9435:
9429:
9426:
9422:
9419:
9416:
9413:
9410:
9404:
9401:
9395:
9392:
9388:
9382:
9379:
9373:
9370:
9365:
9362:
9356:
9353:
9347:
9344:
9294:
9280:
9277:
9274:
9268:
9265:
9262:
9259:
9256:
9253:
9250:
9247:
9221:Thomas Harriot
9192:
9189:
9186:
9183:
9180:
9177:
9174:
9171:
9168:
9163:
9159:
9155:
9152:
9149:
9098:
9095:
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9089:
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9083:
9080:
9077:
9073:
9067:
9063:
9057:
9052:
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9046:
9042:
9037:
9033:
9028:
9024:
9020:
8989:
8938:
8935:
8926:
8923:
8890:: that is use
8876:and the angle
8826:
8823:
8796:
8795:
8781:
8763:
8753:
8727:
8717:
8707:
8676:
8673:
8668:Main article:
8665:
8662:
8649:
8646:
8643:
8639:
8636:
8633:
8629:
8626:
8623:
8620:
8617:
8614:
8611:
8607:
8604:
8601:
8598:
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8589:
8585:
8582:
8579:
8575:
8572:
8569:
8566:
8563:
8560:
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8553:
8550:
8547:
8527:
8524:
8519:
8515:
8510:
8507:
8504:
8500:
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8491:
8488:
8485:
8480:
8476:
8471:
8468:
8465:
8461:
8458:
8455:
8435:
8432:
8429:
8424:
8420:
8415:
8412:
8409:
8405:
8402:
8399:
8396:
8393:
8390:
8387:
8383:
8380:
8377:
8374:
8371:
8368:
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8355:
8351:
8348:
8345:
8342:
8339:
8336:
8333:
8330:
8327:
8324:
8321:
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8288:
8283:
8279:
8274:
8271:
8268:
8264:
8261:
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8230:
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4887:
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4878:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
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4831:
4828:
4825:
4822:
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4778:
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4772:
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4713:
4710:
4703:
4700:
4698:
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4690:
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4675:
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4664:
4661:
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4407:
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4369:
4364:
4361:
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4340:
4337:
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4328:
4325:
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4305:
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4296:
4293:
4290:
4289:
4270:
4267:
4264:
4261:
4258:
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4240:
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4217:
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4208:
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4199:
4196:
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4181:
4178:
4150:
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4144:
4141:
4138:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4111:
4108:
4105:
4102:
4099:
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4093:
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4087:
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4069:
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4057:
4053:
4049:
4046:
4043:
4040:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
4015:
4012:
4009:
4006:
4003:
4000:
3997:
3994:
3991:
3988:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
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3909:
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3900:
3897:
3894:
3891:
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3879:
3876:
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3870:
3868:
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3813:
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3799:
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3789:
3786:
3783:
3780:
3777:
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3764:
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3741:
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3727:
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3714:
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3639:
3636:
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3566:
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3560:
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3534:
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3523:
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3519:
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3493:
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3486:
3483:
3480:
3477:
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3471:
3468:
3464:
3461:
3458:
3455:
3452:
3450:
3448:
3445:
3442:
3438:
3435:
3432:
3429:
3421:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3397:
3394:
3391:
3388:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3362:
3359:
3356:
3353:
3350:
3348:
3346:
3343:
3340:
3336:
3333:
3330:
3327:
3319:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3295:
3291:
3288:
3285:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3259:
3256:
3253:
3250:
3247:
3245:
3243:
3240:
3237:
3233:
3230:
3227:
3223:
3215:
3214:
3194:
3189:
3174:
3173:
3165:
3164:
3158:
3152:
3147:
3132:
3131:
3123:
3122:
3116:
3110:
3107:
3102:
3087:
3086:
3078:
3077:
3071:
3065:
3060:
3045:
3044:
3036:
3035:
3029:
3023:
3020:
3015:
3000:
2999:
2991:
2990:
2984:
2978:
2973:
2958:
2957:
2949:
2948:
2942:
2936:
2888:
2885:
2869:
2866:
2863:
2860:
2856:
2853:
2850:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2824:
2821:
2818:
2815:
2812:
2809:
2807:
2805:
2802:
2799:
2796:
2795:
2792:
2789:
2786:
2783:
2779:
2776:
2773:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2747:
2744:
2741:
2738:
2735:
2732:
2730:
2728:
2725:
2722:
2719:
2718:
2715:
2712:
2709:
2706:
2702:
2699:
2696:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2670:
2667:
2664:
2661:
2658:
2655:
2653:
2651:
2648:
2645:
2642:
2641:
2597:
2594:
2592:
2589:
2444:
2441:
2412:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2338:
2334:
2330:
2327:
2321:
2317:
2313:
2310:
2304:
2300:
2296:
2293:
2287:
2284:
2282:
2277:
2274:
2271:
2266:
2263:
2260:
2254:
2253:
2247:
2241:
2237:
2232:
2226:
2222:
2214:
2210:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2159:
2155:
2151:
2148:
2145:
2142:
2139:
2136:
2131:
2127:
2123:
2120:
2117:
2111:
2108:
2106:
2104:
2099:
2094:
2088:
2085:
2082:
2079:
2076:
2073:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2044:
2041:
2035:
2030:
2027:
2024:
2021:
2019:
2017:
2014:
2009:
2005:
2001:
2000:
1973:
1970:
1965:
1961:
1957:
1954:
1951:
1948:
1945:
1940:
1936:
1920:Main article:
1917:
1914:
1899:
1893:
1890:
1887:
1884:
1881:
1878:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1840:
1837:
1834:
1831:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1731:
1728:
1725:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1674:
1670:
1667:
1660:
1654:
1650:
1647:
1602:
1599:
1596:
1593:
1590:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1539:
1537:
1535:
1529:
1525:
1522:
1515:
1514:
1511:
1508:
1505:
1502:
1498:
1495:
1491:
1488:
1485:
1482:
1479:
1475:
1473:
1471:
1465:
1461:
1458:
1451:
1450:
1447:
1444:
1440:
1437:
1433:
1430:
1427:
1423:
1421:
1419:
1413:
1409:
1406:
1399:
1398:
1236:Main article:
1233:
1230:
1213:
1207:
1204:
1201:
1196:
1193:
1190:
1184:
1178:
1175:
1172:
1167:
1164:
1161:
1155:
1149:
1146:
1143:
1138:
1135:
1132:
1116:The spherical
1110:Main article:
1107:
1104:
1087:
1084:
1079:
1075:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1030:
1027:
1024:
1021:
1018:
998:
995:
992:
989:
986:
983:
980:
950:
947:
944:
941:
938:
935:
932:
929:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
894:
892:
889:
886:
883:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
846:
843:
840:
837:
834:
831:
828:
825:
823:
821:
818:
815:
812:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
778:
775:
772:
769:
766:
763:
760:
757:
754:
752:
750:
747:
744:
741:
740:
725:Main article:
722:
719:
717:
714:
689:
686:
683:
680:
677:
674:
672:
669:
666:
662:
660:
657:
654:
651:
648:
645:
643:
640:
637:
633:
631:
628:
625:
622:
619:
616:
614:
611:
608:
604:
603:
600:
597:
594:
591:
588:
585:
583:
580:
577:
572:
570:
567:
564:
561:
558:
555:
553:
550:
547:
542:
540:
537:
534:
531:
528:
525:
523:
520:
517:
513:
512:
453:polar triangle
434:
431:
430:
429:
402:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
335:
334:
322:
319:
316:
313:
310:
307:
304:
301:
298:
295:
272:
243:
228:
203:
200:
170:, also called
159:plane geometry
132:
129:
119:
116:
26:
9:
6:
4:
3:
2:
11077:
11066:
11063:
11062:
11060:
11051:
11048:
11045:
11042:
11039:
11036:
11031:
11027:
11023:
11020:
11017:
11014:
11011:
11005:
11004:
10999:
10996:
10991:
10985:
10984:
10979:
10976:
10971:
10970:
10950:
10946:
10942:
10936:
10920:
10916:
10915:
10907:
10898:
10897:
10892:
10887:
10881:
10874:
10870:
10867:
10862:
10854:
10850:
10849:
10844:
10840:
10833:
10826:
10820:
10805:
10801:
10800:
10792:
10786:
10782:
10779:
10775:
10762:
10758:
10757:
10752:
10746:
10731:
10727:
10726:
10721:
10715:
10707:
10703:
10699:
10695:
10688:
10674:
10670:
10666:
10662:
10658:
10654:
10650:
10646:
10642:
10638:
10631:
10615:
10614:
10609:
10606:
10599:
10591:
10587:
10586:
10578:
10570:
10566:
10562:
10558:
10557:
10549:
10534:
10530:
10529:
10524:
10523:Todhunter, I.
10518:
10516:
10514:
10512:
10510:
10508:
10506:
10504:
10502:
10500:
10498:
10496:
10494:
10492:
10490:
10488:
10483:
10473:
10470:
10468:
10465:
10463:
10460:
10458:
10455:
10453:
10452:Lenart sphere
10450:
10447:
10444:
10442:
10439:
10437:
10434:
10432:
10429:
10428:
10422:
10420:
10416:
10412:
10408:
10407:line integral
10403:
10384:
10380:
10376:
10371:
10367:
10355:
10351:
10347:
10342:
10338:
10329:
10326:
10321:
10316:
10312:
10289:
10285:
10281:
10276:
10272:
10268:
10263:
10259:
10255:
10250:
10246:
10236:
10223:
10215:
10211:
10207:
10202:
10198:
10188:
10185:
10179:
10176:
10165:
10161:
10157:
10152:
10148:
10138:
10135:
10129:
10126:
10116:
10112:
10108:
10103:
10099:
10089:
10086:
10080:
10077:
10071:
10066:
10062:
10055:
10052:
10046:
10043:
10018:
10014:
10010:
10005:
10001:
9972:
9968:
9964:
9959:
9955:
9931:
9926:
9922:
9899:
9895:
9880:
9878:
9874:
9873:Angle deficit
9870:
9857:
9854:
9848:
9845:
9839:
9836:
9833:
9827:
9824:
9818:
9815:
9812:
9809:
9803:
9800:
9794:
9791:
9781:
9773:
9762:
9755:
9742:
9736:
9733:
9730:
9727:
9722:
9719:
9714:
9711:
9708:
9703:
9700:
9695:
9692:
9689:
9686:
9681:
9678:
9675:
9672:
9667:
9664:
9659:
9656:
9653:
9648:
9645:
9640:
9637:
9631:
9628:
9622:
9619:
9613:
9610:
9590:
9585:
9582:
9577:
9574:
9571:
9568:
9559:
9543:
9540:
9537:
9534:
9531:
9522:
9519:
9513:
9510:
9485:
9482:
9479:
9470:
9467:
9461:
9458:
9451:
9448:
9445:
9436:
9433:
9427:
9424:
9417:
9414:
9411:
9402:
9399:
9393:
9390:
9386:
9380:
9377:
9371:
9368:
9363:
9360:
9354:
9351:
9345:
9342:
9334:
9329:
9326:
9318:
9305:
9292:
9278:
9275:
9272:
9266:
9263:
9260:
9257:
9254:
9251:
9248:
9245:
9237:
9234:
9231:
9222:
9218:
9217:Albert Girard
9214:
9190:
9187:
9184:
9181:
9178:
9175:
9172:
9169:
9166:
9161:
9157:
9153:
9150:
9147:
9126:
9109:
9096:
9093:
9087:
9084:
9081:
9075:
9071:
9065:
9061:
9055:
9050:
9047:
9044:
9040:
9035:
9031:
9026:
9022:
9018:
8982:Consider an
8969:
8957:
8953:
8948:
8944:
8934:
8922:
8911:
8888:
8881:
8858:
8849:at the point
8819:
8815:
8813:
8808:
8806:
8785:
8782:
8767:
8764:
8757:
8754:
8731:
8728:
8721:
8718:
8711:
8708:
8693:
8690:
8689:
8685:
8681:
8671:
8661:
8647:
8644:
8641:
8637:
8634:
8631:
8627:
8624:
8621:
8618:
8615:
8612:
8609:
8605:
8602:
8599:
8596:
8593:
8590:
8587:
8583:
8580:
8577:
8573:
8570:
8567:
8564:
8561:
8558:
8555:
8551:
8548:
8545:
8525:
8522:
8517:
8513:
8508:
8505:
8502:
8498:
8495:
8492:
8489:
8486:
8483:
8478:
8474:
8469:
8466:
8463:
8459:
8456:
8453:
8433:
8430:
8427:
8422:
8418:
8413:
8410:
8407:
8403:
8400:
8397:
8394:
8391:
8388:
8385:
8381:
8378:
8375:
8372:
8369:
8366:
8363:
8359:
8356:
8353:
8349:
8346:
8343:
8340:
8337:
8334:
8331:
8328:
8325:
8322:
8319:
8310:
8292:
8289:
8286:
8281:
8277:
8272:
8269:
8266:
8262:
8259:
8256:
8253:
8250:
8247:
8244:
8240:
8237:
8234:
8231:
8228:
8225:
8222:
8218:
8215:
8212:
8208:
8205:
8202:
8199:
8196:
8193:
8190:
8187:
8184:
8181:
8172:
8160:
8157:
8144:
8141:
8138:
8134:
8131:
8128:
8125:
8122:
8119:
8116:
8112:
8109:
8106:
8102:
8099:
8096:
8093:
8090:
8087:
8084:
8081:
8078:
8075:
8066:
8044:
8041:
8038:
8034:
8031:
8028:
8024:
8021:
8018:
8015:
8012:
8009:
8006:
8002:
7999:
7996:
7992:
7989:
7986:
7982:
7979:
7976:
7973:
7971:
7966:
7963:
7958:
7954:
7949:
7946:
7943:
7936:
7933:
7930:
7926:
7923:
7920:
7916:
7913:
7910:
7907:
7904:
7901:
7898:
7892:
7889:
7886:
7882:
7879:
7876:
7872:
7869:
7866:
7863:
7860:
7857:
7854:
7850:
7847:
7844:
7838:
7836:
7831:
7828:
7825:
7807:
7790:
7787:
7784:
7781:
7777:
7774:
7771:
7768:
7765:
7763:
7758:
7755:
7752:
7739:
7736:
7733:
7730:
7726:
7723:
7720:
7717:
7715:
7710:
7707:
7704:
7689:
7686:
7683:
7680:
7676:
7673:
7670:
7667:
7665:
7660:
7657:
7654:
7641:
7638:
7635:
7632:
7628:
7625:
7622:
7619:
7617:
7612:
7609:
7606:
7591:
7588:
7585:
7582:
7578:
7575:
7572:
7569:
7567:
7562:
7559:
7556:
7543:
7540:
7537:
7534:
7530:
7527:
7524:
7521:
7519:
7514:
7511:
7508:
7493:
7490:
7487:
7484:
7480:
7477:
7474:
7471:
7468:
7466:
7461:
7458:
7455:
7442:
7439:
7436:
7433:
7429:
7426:
7423:
7420:
7418:
7413:
7410:
7407:
7392:
7389:
7386:
7383:
7379:
7376:
7373:
7370:
7367:
7365:
7360:
7357:
7354:
7337:
7334:
7331:
7328:
7324:
7321:
7318:
7315:
7312:
7310:
7305:
7302:
7299:
7277:
7269:
7263:
7255:
7245:
7218:
7209:
7192:
7189:
7186:
7183:
7179:
7176:
7173:
7170:
7168:
7163:
7160:
7157:
7144:
7141:
7138:
7135:
7131:
7128:
7125:
7122:
7120:
7115:
7112:
7109:
7094:
7091:
7088:
7085:
7081:
7078:
7075:
7072:
7070:
7065:
7062:
7059:
7046:
7043:
7040:
7037:
7033:
7030:
7027:
7024:
7022:
7017:
7014:
7011:
6996:
6993:
6990:
6987:
6983:
6980:
6977:
6974:
6972:
6967:
6964:
6961:
6948:
6945:
6942:
6939:
6935:
6932:
6929:
6926:
6924:
6919:
6916:
6913:
6898:
6895:
6892:
6889:
6885:
6882:
6879:
6876:
6874:
6869:
6866:
6863:
6850:
6847:
6844:
6841:
6837:
6834:
6831:
6828:
6826:
6821:
6818:
6815:
6800:
6797:
6794:
6791:
6787:
6784:
6781:
6778:
6776:
6771:
6768:
6765:
6748:
6745:
6742:
6739:
6735:
6732:
6729:
6726:
6724:
6719:
6716:
6713:
6676:
6673:
6670:
6667:
6663:
6660:
6657:
6654:
6652:
6644:
6641:
6638:
6634:
6631:
6628:
6625:
6623:
6612:
6609:
6603:
6600:
6591:
6588:
6581:
6578:
6572:
6569:
6560:
6557:
6554:
6552:
6544:
6541:
6538:
6531:
6528:
6522:
6519:
6510:
6507:
6504:
6502:
6497:
6494:
6491:
6470:
6467:
6466:
6465:
6463:
6429:
6427:
6423:
6419:
6385:
6376:
6357:
6354:
6351:
6342:
6339:
6333:
6330:
6322:
6319:
6316:
6307:
6304:
6298:
6295:
6289:
6280:
6277:
6274:
6265:
6262:
6256:
6253:
6245:
6242:
6239:
6230:
6227:
6221:
6218:
6207:
6206:(1201β1274),
6205:
6202:
6198:
6193:
6190:
6187:
6170:
6164:
6161:
6155:
6152:
6143:
6140:
6137:
6128:
6125:
6119:
6116:
6108:
6105:
6102:
6093:
6090:
6084:
6081:
6075:
6069:
6066:
6063:
6054:
6051:
6045:
6042:
6034:
6028:
6025:
6019:
6016:
6007:
6004:
6001:
5992:
5989:
5983:
5980:
5972:
5969:
5966:
5957:
5954:
5948:
5945:
5939:
5933:
5930:
5927:
5918:
5915:
5909:
5906:
5899:
5893:
5890:
5884:
5881:
5872:
5869:
5866:
5857:
5854:
5848:
5845:
5837:
5834:
5831:
5822:
5819:
5813:
5810:
5804:
5798:
5795:
5792:
5783:
5780:
5774:
5771:
5763:
5757:
5754:
5748:
5745:
5736:
5733:
5730:
5721:
5718:
5712:
5709:
5701:
5698:
5695:
5686:
5683:
5677:
5674:
5668:
5662:
5659:
5656:
5647:
5644:
5638:
5635:
5618:
5615:
5594:
5588:
5585:
5579:
5576:
5568:
5565:
5562:
5553:
5550:
5544:
5541:
5535:
5529:
5523:
5520:
5514:
5511:
5503:
5500:
5497:
5488:
5485:
5479:
5476:
5462:
5456:
5453:
5447:
5444:
5436:
5433:
5430:
5421:
5418:
5412:
5409:
5403:
5397:
5391:
5388:
5382:
5379:
5371:
5368:
5365:
5356:
5353:
5347:
5344:
5331:
5325:
5322:
5316:
5313:
5305:
5302:
5299:
5290:
5287:
5281:
5278:
5272:
5266:
5260:
5257:
5251:
5248:
5240:
5237:
5234:
5225:
5222:
5216:
5213:
5198:
5192:
5189:
5183:
5180:
5172:
5169:
5166:
5157:
5154:
5148:
5145:
5139:
5133:
5127:
5124:
5118:
5115:
5107:
5104:
5101:
5092:
5089:
5083:
5080:
5065:
5057:
5043:
5040:
5037:
5034:
5031:
5028:
5025:
5019:
5016:
5007:
5003:
4999:
4991:
4971:
4968:
4965:
4962:
4959:
4956:
4953:
4947:
4944:
4935:
4931:
4927:
4918:
4915:
4891:
4888:
4885:
4879:
4876:
4870:
4867:
4864:
4858:
4855:
4847:
4844:
4841:
4835:
4832:
4829:
4826:
4823:
4820:
4813:
4811:
4806:
4800:
4797:
4791:
4788:
4776:
4773:
4770:
4764:
4761:
4758:
4755:
4752:
4744:
4741:
4738:
4732:
4729:
4723:
4720:
4717:
4711:
4708:
4701:
4699:
4694:
4688:
4685:
4679:
4676:
4665:
4662:
4659:
4656:
4653:
4650:
4642:
4639:
4636:
4630:
4627:
4621:
4618:
4615:
4609:
4606:
4599:
4597:
4592:
4586:
4583:
4577:
4574:
4565:
4562:
4559:
4556:
4553:
4550:
4542:
4539:
4536:
4530:
4527:
4524:
4521:
4518:
4511:
4509:
4504:
4498:
4495:
4489:
4486:
4475:
4472:
4469:
4466:
4463:
4460:
4452:
4449:
4446:
4440:
4437:
4434:
4431:
4428:
4425:
4418:
4416:
4411:
4405:
4402:
4396:
4393:
4382:
4379:
4376:
4373:
4370:
4367:
4359:
4356:
4353:
4347:
4344:
4338:
4335:
4332:
4326:
4323:
4316:
4314:
4309:
4303:
4300:
4294:
4291:
4268:
4262:
4259:
4256:
4253:
4250:
4244:
4241:
4238:
4215:
4212:
4209:
4206:
4203:
4197:
4194:
4191:
4177:
4174:
4170:
4148:
4145:
4142:
4139:
4136:
4133:
4130:
4127:
4124:
4121:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4068:
4063:
4060:
4055:
4051:
4047:
4044:
4041:
4034:
4031:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
4007:
4004:
4001:
3998:
3992:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3938:
3935:
3932:
3929:
3927:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3889:
3886:
3883:
3880:
3877:
3874:
3871:
3869:
3864:
3861:
3858:
3845:
3820:
3817:
3814:
3811:
3806:
3800:
3797:
3794:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3768:
3765:
3762:
3759:
3757:
3752:
3749:
3746:
3742:
3739:
3736:
3718:
3715:
3712:
3709:
3704:
3698:
3695:
3692:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3666:
3663:
3660:
3657:
3655:
3650:
3647:
3644:
3640:
3637:
3634:
3616:
3613:
3610:
3607:
3602:
3596:
3593:
3590:
3586:
3583:
3580:
3577:
3574:
3571:
3568:
3564:
3561:
3558:
3555:
3553:
3548:
3545:
3542:
3538:
3535:
3532:
3514:
3511:
3508:
3505:
3500:
3494:
3491:
3488:
3484:
3481:
3478:
3475:
3472:
3469:
3466:
3462:
3459:
3456:
3453:
3451:
3446:
3443:
3440:
3436:
3433:
3430:
3412:
3409:
3406:
3403:
3398:
3392:
3389:
3386:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3360:
3357:
3354:
3351:
3349:
3344:
3341:
3338:
3334:
3331:
3328:
3310:
3307:
3304:
3301:
3296:
3289:
3286:
3283:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3257:
3254:
3251:
3248:
3246:
3241:
3238:
3235:
3231:
3228:
3225:
3192:
3150:
3108:
3105:
3063:
3021:
3018:
2976:
2934:
2898:
2884:
2867:
2864:
2861:
2858:
2854:
2851:
2848:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2822:
2819:
2816:
2813:
2810:
2808:
2803:
2800:
2797:
2790:
2787:
2784:
2781:
2777:
2774:
2771:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2745:
2742:
2739:
2736:
2733:
2731:
2726:
2723:
2720:
2713:
2710:
2707:
2704:
2700:
2697:
2694:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2668:
2665:
2662:
2659:
2656:
2654:
2649:
2646:
2643:
2630:
2616:
2603:
2588:
2586:
2581:
2577:
2573:
2569:
2562:
2558:
2554:
2546:
2537:
2528:
2504:
2500:
2496:
2491:evaluates to
2488:
2479:
2470:
2460:
2454:
2451:
2440:
2410:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2336:
2332:
2328:
2325:
2319:
2315:
2311:
2308:
2302:
2298:
2294:
2291:
2285:
2283:
2275:
2272:
2269:
2264:
2261:
2258:
2245:
2239:
2235:
2230:
2224:
2220:
2212:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2171:
2165:
2162:
2157:
2153:
2149:
2146:
2137:
2134:
2129:
2125:
2121:
2118:
2109:
2107:
2097:
2092:
2086:
2083:
2080:
2077:
2074:
2071:
2066:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2042:
2039:
2033:
2028:
2025:
2022:
2020:
2015:
2012:
2007:
2003:
1989:
1971:
1968:
1963:
1959:
1955:
1952:
1949:
1946:
1943:
1938:
1934:
1923:
1913:
1910:
1897:
1891:
1888:
1885:
1882:
1879:
1876:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1838:
1835:
1832:
1829:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1668:
1665:
1658:
1648:
1645:
1617:
1600:
1594:
1591:
1588:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1538:
1533:
1523:
1520:
1506:
1503:
1500:
1496:
1493:
1489:
1486:
1483:
1480:
1474:
1469:
1459:
1456:
1442:
1438:
1435:
1431:
1428:
1422:
1417:
1407:
1404:
1388:
1322:
1318:
1304:
1288:
1279:
1270:
1259:
1257:
1253:
1244:
1239:
1229:
1227:
1211:
1205:
1202:
1199:
1194:
1191:
1188:
1182:
1176:
1173:
1170:
1165:
1162:
1159:
1153:
1147:
1144:
1141:
1136:
1133:
1130:
1119:
1113:
1103:
1101:
1085:
1082:
1077:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1028:
1025:
1022:
1019:
1016:
996:
990:
987:
984:
981:
978:
970:
965:
948:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
895:
890:
887:
884:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
832:
829:
826:
824:
819:
816:
813:
806:
803:
800:
797:
794:
791:
788:
785:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
753:
748:
745:
742:
728:
713:
709:
687:
684:
681:
678:
675:
673:
667:
664:
658:
655:
652:
649:
646:
644:
638:
635:
629:
626:
623:
620:
617:
615:
609:
606:
598:
595:
592:
589:
586:
584:
578:
575:
568:
565:
562:
559:
556:
554:
548:
545:
538:
535:
532:
529:
526:
524:
518:
515:
501:
494:
489:The triangle
487:
478:. The points
460:
454:
446:
439:
403:
388:
385:
382:
379:
376:
373:
370:
367:
364:
361:
349:
344:
343:
342:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
281:
277:
273:
270:
263:
248:
244:
229:
214:
213:
208:
199:
197:
192:
190:
186:
181:
179:
175:
174:
169:
168:
162:
160:
156:
155:line segments
152:
151:great circles
148:
144:
143:
138:
124:
118:Preliminaries
115:
113:
109:
105:
101:
97:
92:
90:
86:
82:
78:
77:great circles
74:
70:
66:
62:
58:
54:
50:
46:
39:
34:
30:
19:
11034:by Okay Arik
11001:
10981:
10953:. Retrieved
10944:
10935:
10923:. Retrieved
10913:
10906:
10895:
10880:
10861:
10846:
10832:
10824:
10819:
10808:. Retrieved
10798:
10791:
10773:
10765:. Retrieved
10755:
10745:
10734:. Retrieved
10724:
10714:
10697:
10693:
10687:
10677:, retrieved
10644:
10640:
10630:
10617:. Retrieved
10611:
10598:
10584:
10577:
10555:
10548:
10537:. Retrieved
10527:
10413:, or via an
10404:
10237:
9886:
9871:
9779:
9771:
9760:
9756:
9560:
9330:
9324:
9316:
9306:
9238:
9235:
9229:
9212:
9124:
9110:
8981:
8968:small circle
8928:
8913:. The angle
8909:
8886:
8879:
8856:
8828:
8809:
8797:
8783:
8765:
8755:
8729:
8719:
8709:
8691:
8678:
8308:
8170:
8166:
8158:
8064:
7813:
7275:
7267:
7261:
7253:
7243:
7223:
6474:
6461:
6430:
6426:mnemonic aid
6420:
6389:
6208:
6194:
6191:
6188:
5624:
5616:
5066:
5063:
4919:
4916:
4183:
4172:
4168:
3843:
2896:
2890:
2628:
2614:
2601:
2599:
2585:law of sines
2579:
2575:
2571:
2567:
2560:
2556:
2552:
2539:
2530:
2521:
2510:-axis along
2502:
2498:
2494:
2481:
2472:
2463:
2455:
2446:
1987:
1925:
1911:
1618:
1389:
1367:projects to
1320:
1311:
1302:
1281:
1272:
1263:
1260:
1248:
1226:trigonometry
1118:law of sines
1115:
969:trigonometry
966:
730:
721:Cosine rules
707:
499:
492:
488:
458:
452:
450:
444:
347:
336:
279:
261:
246:
195:
193:
189:star polygon
182:
177:
171:
165:
163:
140:
136:
134:
111:
93:
60:
44:
43:
29:
8993:denote the
8943:Solid angle
7247:with sides
7236:has length
2897:consecutive
104:John Napier
10955:2021-05-01
10810:2021-07-11
10767:2016-05-14
10736:2016-05-14
10679:2016-01-10
10539:2013-07-28
10478:References
8941:See also:
7251:such that
7249:a', b', c'
2604:replacing
2591:Identities
1337:-axis and
1333:along the
1106:Sine rules
1098:etc.; see
354:, so that
286:, so that
89:navigation
11003:MathWorld
10983:MathWorld
10778:Abe Books
10751:Napier, J
10669:122277398
10613:MathWorld
10421:formula.
10381:λ
10377:−
10368:λ
10352:φ
10339:φ
10322:≈
10286:λ
10282:−
10273:λ
10260:φ
10247:φ
10212:λ
10208:−
10199:λ
10180:
10162:φ
10158:−
10149:φ
10130:
10113:φ
10100:φ
10081:
10047:
10015:φ
10002:λ
9969:φ
9956:λ
9923:λ
9896:λ
9840:
9819:
9795:
9734:
9715:
9696:
9679:
9660:
9641:
9614:
9578:≈
9483:−
9462:
9449:−
9428:
9415:−
9394:
9372:
9346:
9333:L'Huilier
9279:×
9276:π
9264:π
9188:π
9185:−
9148:≡
9094:π
9085:−
9076:−
9041:∑
9019:≡
8917:and side
8645:
8635:
8625:
8619:−
8613:
8603:
8591:
8581:
8571:
8559:
8549:
8523:
8506:
8496:
8484:
8467:
8457:
8428:
8411:
8401:
8395:−
8389:
8379:
8367:
8357:
8347:
8341:−
8332:
8323:
8287:
8270:
8260:
8254:−
8248:
8238:
8226:
8216:
8206:
8194:
8185:
8142:
8132:
8120:
8110:
8100:
8088:
8079:
8042:
8032:
8022:
8010:
8000:
7990:
7980:
7964:
7947:
7934:
7924:
7914:
7902:
7890:
7880:
7870:
7858:
7848:
7829:
7785:
7775:
7769:−
7756:
7734:
7724:
7708:
7684:
7674:
7658:
7636:
7626:
7610:
7586:
7576:
7560:
7538:
7528:
7512:
7488:
7478:
7472:−
7459:
7437:
7427:
7411:
7387:
7377:
7371:−
7358:
7332:
7322:
7316:−
7303:
7187:
7177:
7161:
7139:
7129:
7113:
7089:
7079:
7063:
7041:
7031:
7015:
6991:
6981:
6965:
6943:
6933:
6917:
6893:
6883:
6867:
6845:
6835:
6819:
6795:
6785:
6769:
6743:
6733:
6717:
6671:
6661:
6642:
6632:
6610:−
6601:π
6592:
6579:−
6570:π
6561:
6542:
6529:−
6520:π
6511:
6495:
6480:we have:
6444:(that is
6334:
6320:−
6299:
6257:
6243:−
6222:
6156:
6120:
6106:−
6085:
6067:−
6046:
6020:
5984:
5970:−
5949:
5931:−
5910:
5885:
5849:
5835:−
5814:
5775:
5749:
5713:
5699:−
5678:
5639:
5580:
5545:
5515:
5501:−
5480:
5448:
5413:
5383:
5348:
5317:
5303:−
5282:
5252:
5238:−
5217:
5184:
5170:−
5149:
5119:
5084:
5038:
4966:
4960:−
4889:−
4880:
4868:−
4859:
4845:−
4836:
4827:
4821:−
4792:
4774:−
4765:
4756:
4742:−
4733:
4721:−
4712:
4680:
4663:
4654:
4640:−
4631:
4619:−
4610:
4578:
4563:
4554:
4540:−
4531:
4522:
4490:
4473:
4464:
4450:−
4441:
4432:
4426:−
4397:
4380:
4371:
4357:−
4348:
4336:−
4327:
4295:
4143:
4134:
4125:
4116:
4104:
4095:
4086:
4077:
4061:
4045:
4032:
4023:
4014:
4005:
3990:
3981:
3972:
3960:
3951:
3936:
3917:
3908:
3899:
3887:
3878:
3862:
3798:
3788:
3782:−
3776:
3766:
3750:
3740:
3696:
3686:
3680:−
3674:
3664:
3648:
3638:
3594:
3584:
3578:−
3572:
3562:
3546:
3536:
3492:
3482:
3476:−
3470:
3460:
3444:
3434:
3390:
3380:
3374:−
3368:
3358:
3342:
3332:
3287:
3277:
3271:−
3265:
3255:
3239:
3229:
3106:−
2862:
2852:
2842:
2830:
2820:
2814:−
2801:
2785:
2775:
2765:
2753:
2743:
2737:−
2724:
2708:
2698:
2688:
2676:
2666:
2660:−
2647:
2402:
2393:
2384:
2373:
2364:
2355:
2329:−
2312:−
2295:−
2273:
2262:
2202:
2193:
2187:−
2181:
2172:−
2163:
2150:−
2135:
2122:−
2084:
2075:
2064:
2055:
2049:−
2043:
2029:−
2013:
1969:
1956:−
1944:
1889:
1880:
1869:
1860:
1854:−
1848:
1833:
1804:
1795:
1786:
1774:
1765:
1753:
1724:
1715:
1703:
1694:
1685:
1673:→
1659:⋅
1653:→
1592:
1579:
1570:
1557:
1548:
1528:→
1504:
1484:
1464:→
1412:→
1383:-axis is
1354:with the
1203:
1192:
1174:
1163:
1145:
1134:
1083:≈
1067:
1061:−
1055:
1026:≈
1020:
994:→
943:
934:
925:
913:
904:
888:
872:
863:
854:
842:
833:
817:
801:
792:
783:
771:
762:
746:
682:−
679:π
653:−
650:π
624:−
621:π
593:−
590:π
563:−
560:π
533:−
530:π
389:π
341:radians.
321:π
294:π
196:triangles
178:bi-angles
81:astronomy
73:geodesics
67:. On the
11059:Category
10949:Archived
10925:7 August
10919:Archived
10893:(1880).
10869:Archived
10804:Archived
10781:Archived
10761:Archived
10753:(1614).
10730:Archived
10722:(1807).
10673:archived
10533:Archived
10525:(1886).
10425:See also
8966:along a
1379:and the
668:′
639:′
610:′
579:′
549:′
519:′
269:tangents
202:Notation
108:Delambre
10896:Geodesy
10661:4146847
10619:8 April
10565:2484948
10556:Geodesy
9768:, then
8748:, then
8740:) give
8736:) and (
8312:yields
7244:A'B'C'
1371:in the
1346:in the
493:A'B'C'
445:A'B'C'
276:radians
142:polygon
85:geodesy
11013:TriSph
10667:
10659:
10563:
9503:where
9317:proper
9203:where
9119:, and
8860:: use
8837:, and
8776:, and
8702:, and
8174:gives
8068:gives
6462:middle
6452:, and
6438:aCbAcB
6436:gives
6422:Napier
6414:, and
3942:
2893:aCbAcB
2632:etc.,
2574:= sin
2435:, and
1319:= cos
1252:vector
420:, and
348:proper
280:proper
262:vertex
238:, and
223:, and
173:digons
87:, and
69:sphere
57:angles
10665:S2CID
10657:JSTOR
10409:with
9914:and
7747:(Q10)
7152:(R10)
4184:With
3730:(CT6)
3628:(CT5)
3526:(CT4)
3424:(CT3)
3322:(CT2)
3218:(CT1)
3177:angle
3168:inner
3135:angle
3126:outer
3081:inner
3039:outer
3003:angle
2994:inner
2952:inner
2903:) or
247:angle
167:lunes
139:is a
53:sides
10927:2020
10621:2018
10561:OCLC
9990:and
9778:sin
9776:and
9770:cos
9315:for
8974:and
8945:and
8906:and
8894:and
8872:and
8864:and
8790:and
8744:and
8738:BaCb
8734:cBaC
8307:cos
8169:cos
8063:sin
7699:(Q5)
7649:(Q9)
7601:(Q4)
7551:(Q8)
7503:(Q3)
7450:(Q7)
7402:(Q2)
7348:(Q6)
7293:(Q1)
7104:(R5)
7054:(R9)
7006:(R4)
6956:(R8)
6908:(R3)
6858:(R7)
6810:(R2)
6759:(R6)
6707:(R1)
4231:and
4171:sin
4167:sin
3842:cos
3090:side
3048:side
2961:side
2909:BaCb
2905:BaCb
2901:aCbA
2602:i.e.
2578:sin
2570:sin
2566:sin
2559:sin
2555:sin
2551:sin
2501:sin
2497:sin
2493:sin
2457:The
1986:cos
1627:Β· OC
1041:and
1009:set
482:and
451:The
383:<
365:<
315:<
297:<
256:and
245:The
147:arcs
98:and
75:are
55:and
36:The
10702:doi
10649:doi
10177:tan
10127:cos
10078:sin
10044:tan
10036:is
9837:tan
9816:tan
9792:tan
9782:= 1
9774:= 0
9761:ABC
9731:cos
9712:tan
9693:tan
9676:sin
9657:tan
9638:tan
9611:tan
9459:tan
9425:tan
9391:tan
9369:tan
9343:tan
8910:DAC
8887:ACD
8880:BAD
8857:ABD
8642:cos
8632:cos
8622:cos
8610:sin
8600:sin
8588:cos
8578:cos
8568:cos
8556:sin
8546:sin
8514:sin
8503:sin
8493:sin
8475:sin
8464:sin
8454:sin
8419:sin
8408:sin
8398:sin
8386:sin
8376:sin
8364:cos
8354:cos
8344:cos
8329:cos
8320:cos
8278:sin
8267:sin
8257:sin
8245:sin
8235:sin
8223:cos
8213:cos
8203:cos
8191:cos
8182:cos
8139:cos
8129:sin
8117:cos
8107:cos
8097:sin
8085:sin
8076:cos
8039:cos
8029:sin
8019:sin
8007:cos
7997:sin
7987:cos
7977:sin
7955:sin
7944:cos
7931:cos
7921:sin
7911:sin
7899:cos
7887:cos
7877:sin
7867:sin
7855:cos
7845:cos
7826:cos
7782:cot
7772:cot
7753:cos
7731:sin
7721:tan
7705:tan
7681:cos
7671:sin
7655:cos
7633:sin
7623:tan
7607:tan
7583:cos
7573:sin
7557:cos
7535:sin
7525:sin
7509:sin
7485:tan
7475:cos
7456:tan
7434:sin
7424:sin
7408:sin
7384:tan
7374:cos
7355:tan
7329:cos
7319:cos
7300:cos
7268:a'
7254:A'
7184:cot
7174:cot
7158:cos
7136:sin
7126:tan
7110:tan
7086:cos
7076:sin
7060:cos
7038:sin
7028:tan
7012:tan
6988:cos
6978:sin
6962:cos
6940:sin
6930:sin
6914:sin
6890:tan
6880:cos
6864:tan
6842:sin
6832:sin
6816:sin
6792:tan
6782:cos
6766:tan
6740:cos
6730:cos
6714:cos
6668:sin
6658:sin
6639:tan
6629:cot
6589:cos
6558:cos
6539:tan
6508:tan
6492:sin
6331:tan
6296:tan
6254:tan
6219:tan
6153:tan
6117:sin
6082:sin
6043:tan
6017:cot
5981:sin
5946:sin
5907:tan
5882:tan
5846:cos
5811:cos
5772:tan
5746:cot
5710:cos
5675:cos
5636:tan
5577:sin
5542:sin
5512:sin
5477:cos
5445:cos
5410:cos
5380:sin
5345:cos
5314:sin
5279:sin
5249:cos
5214:sin
5181:cos
5146:cos
5116:cos
5081:sin
5035:cos
5004:cos
4963:cos
4932:sin
4877:cos
4856:cos
4833:cos
4824:cos
4789:tan
4762:sin
4753:sin
4730:sin
4709:sin
4677:tan
4660:sin
4651:sin
4628:cos
4607:cos
4575:cos
4560:sin
4551:sin
4528:sin
4519:sin
4487:cos
4470:sin
4461:sin
4438:cos
4429:cos
4394:sin
4377:sin
4368:sin
4345:sin
4324:sin
4292:sin
4140:cot
4131:sin
4122:sin
4113:sin
4101:cos
4092:sin
4083:sin
4074:cos
4052:sin
4042:cos
4029:cot
4020:sin
4011:sin
4002:sin
3987:cos
3978:sin
3969:sin
3957:cos
3948:cos
3933:cos
3914:cos
3905:sin
3896:sin
3884:cos
3875:cos
3859:cos
3795:sin
3785:cot
3773:sin
3763:cot
3747:cos
3737:cos
3693:sin
3683:cot
3671:sin
3661:cot
3645:cos
3635:cos
3591:sin
3581:cot
3569:sin
3559:cot
3543:cos
3533:cos
3489:sin
3479:cot
3467:sin
3457:cot
3441:cos
3431:cos
3387:sin
3377:cot
3365:sin
3355:cot
3339:cos
3329:cos
3284:sin
3274:cot
3262:sin
3252:cot
3236:cos
3226:cos
3151:sin
3109:cot
3064:sin
3022:cot
2977:cos
2935:cos
2859:cos
2849:sin
2839:sin
2827:cos
2817:cos
2798:cos
2782:cos
2772:sin
2762:sin
2750:cos
2740:cos
2721:cos
2705:cos
2695:sin
2685:sin
2673:cos
2663:cos
2644:cos
2622:by
2608:by
2529:Β· (
2471:Β· (
2399:sin
2390:sin
2381:sin
2370:cos
2361:cos
2352:cos
2333:cos
2316:cos
2299:cos
2270:sin
2259:sin
2236:sin
2221:sin
2199:cos
2190:cos
2178:cos
2154:cos
2126:cos
2081:sin
2072:sin
2061:cos
2052:cos
2040:cos
2004:sin
1960:cos
1935:sin
1886:sin
1877:sin
1866:cos
1857:cos
1845:cos
1830:cos
1801:cos
1792:sin
1783:sin
1771:cos
1762:cos
1750:cos
1721:cos
1712:cos
1700:cos
1691:sin
1682:sin
1589:cos
1576:sin
1567:sin
1554:cos
1545:sin
1501:cos
1481:sin
1258:.)
1200:sin
1189:sin
1171:sin
1160:sin
1142:sin
1131:sin
1102:.)
1064:cos
1052:cos
1017:sin
940:cos
931:sin
922:sin
910:cos
901:cos
885:cos
869:cos
860:sin
851:sin
839:cos
830:cos
814:cos
798:cos
789:sin
780:sin
768:cos
759:cos
743:cos
708:ABC
500:ABC
459:ABC
176:or
157:in
149:of
59:of
11061::
11028:.
11000:.
10980:.
10947:.
10943:.
10851:,
10845:,
10841:,
10698:45
10696:.
10671:,
10663:,
10655:,
10645:35
10643:,
10639:,
10610:.
10486:^
10402:.
9879:.
9335::
9325:is
9311:(3
9115:,
8960:AB
8900:DC
8892:AD
8874:BD
8870:AD
8847:BC
8833:,
8772:,
8698:,
7274:β
7270:=
7265:,
7260:β
7256:=
6448:,
6418:.
6410:,
6406:,
6402:,
2627:β
2618:,
2613:β
2540:OA
2538:Γ
2531:OC
2522:OB
2512:OB
2482:OC
2480:Γ
2473:OB
2464:OA
2461:,
2431:,
1621:OB
1377:ON
1373:xy
1369:ON
1360:OC
1348:xz
1339:OB
1326:OA
1312:OC
1310:Β·
1303:OB
1293:BC
1282:OC
1280:,
1273:OB
1271:,
1264:OA
484:C'
480:B'
476:A'
464:BC
416:,
234:,
219:,
198:.
161:.
135:A
106:,
91:.
83:,
71:,
11032:.
11006:.
10986:.
10958:.
10929:.
10875:.
10813:.
10770:.
10739:.
10708:.
10704::
10651::
10623:.
10592:.
10571:.
10542:.
10390:)
10385:1
10372:2
10364:(
10361:)
10356:1
10348:+
10343:2
10335:(
10330:2
10327:1
10317:4
10313:E
10290:1
10277:2
10269:,
10264:2
10256:,
10251:1
10224:.
10221:)
10216:1
10203:2
10195:(
10189:2
10186:1
10171:)
10166:1
10153:2
10145:(
10139:2
10136:1
10122:)
10117:1
10109:+
10104:2
10096:(
10090:2
10087:1
10072:=
10067:4
10063:E
10056:2
10053:1
10024:)
10019:2
10011:,
10006:2
9998:(
9978:)
9973:1
9965:,
9960:1
9952:(
9932:,
9927:2
9900:1
9858:.
9855:b
9849:2
9846:1
9834:a
9828:2
9825:1
9813:=
9810:E
9804:2
9801:1
9780:C
9772:C
9766:C
9759:β³
9743:.
9737:C
9728:b
9723:2
9720:1
9709:a
9704:2
9701:1
9690:+
9687:1
9682:C
9673:b
9668:2
9665:1
9654:a
9649:2
9646:1
9632:=
9629:E
9623:2
9620:1
9591:c
9586:2
9583:1
9575:b
9572:=
9569:a
9547:)
9544:c
9541:+
9538:b
9535:+
9532:a
9529:(
9523:2
9520:1
9514:=
9511:s
9489:)
9486:c
9480:s
9477:(
9471:2
9468:1
9455:)
9452:b
9446:s
9443:(
9437:2
9434:1
9421:)
9418:a
9412:s
9409:(
9403:2
9400:1
9387:s
9381:2
9378:1
9364:=
9361:E
9355:4
9352:1
9321:Ο
9313:Ο
9309:Ο
9293:.
9273:4
9267:+
9261:=
9258:C
9255:+
9252:B
9249:+
9246:A
9230:R
9225:R
9209:Ο
9205:E
9191:,
9182:C
9179:+
9176:B
9173:+
9170:A
9167:=
9162:3
9158:E
9154:=
9151:E
9121:C
9117:B
9113:A
9097:.
9091:)
9088:2
9082:N
9079:(
9072:)
9066:n
9062:A
9056:N
9051:1
9048:=
9045:n
9036:(
9032:=
9027:N
9023:E
8995:n
8990:n
8988:A
8984:N
8978:.
8976:B
8972:A
8964:C
8931:Ο
8919:a
8915:A
8908:β
8904:C
8896:b
8885:β³
8878:β
8866:B
8862:c
8855:β³
8851:D
8843:A
8839:B
8835:c
8831:b
8801:Ο
8792:A
8788:a
8778:c
8774:b
8770:a
8760:b
8750:A
8746:b
8742:c
8724:C
8714:a
8704:C
8700:B
8696:A
8648:a
8638:C
8628:B
8616:C
8606:B
8597:=
8594:A
8584:c
8574:b
8565:+
8562:c
8552:b
8526:a
8518:2
8509:C
8499:B
8490:=
8487:A
8479:2
8470:c
8460:b
8434:.
8431:a
8423:2
8414:C
8404:B
8392:C
8382:B
8373:+
8370:a
8360:C
8350:B
8338:=
8335:A
8326:a
8309:a
8293:.
8290:A
8282:2
8273:c
8263:b
8251:c
8241:b
8232:+
8229:A
8219:c
8209:b
8200:=
8197:A
8188:a
8171:A
8145:A
8135:b
8126:+
8123:B
8113:c
8103:a
8094:=
8091:c
8082:a
8065:c
8045:A
8035:c
8025:b
8016:+
8013:B
8003:c
7993:c
7983:a
7974:=
7967:c
7959:2
7950:a
7937:A
7927:c
7917:b
7908:+
7905:c
7896:)
7893:B
7883:c
7873:a
7864:+
7861:c
7851:a
7842:(
7839:=
7832:a
7791:.
7788:b
7778:a
7766:=
7759:C
7740:,
7737:A
7727:b
7718:=
7711:B
7690:,
7687:B
7677:a
7668:=
7661:b
7642:,
7639:B
7629:a
7620:=
7613:A
7592:,
7589:A
7579:b
7570:=
7563:a
7544:,
7541:C
7531:b
7522:=
7515:B
7494:,
7491:C
7481:b
7469:=
7462:A
7443:,
7440:C
7430:a
7421:=
7414:A
7393:,
7390:C
7380:a
7368:=
7361:B
7338:,
7335:B
7325:A
7313:=
7306:C
7276:A
7272:Ο
7262:a
7258:Ο
7242:β³
7238:Ο
7234:c
7230:Ο
7226:Ο
7193:.
7190:B
7180:A
7171:=
7164:c
7145:,
7142:a
7132:B
7123:=
7116:b
7095:,
7092:b
7082:A
7073:=
7066:B
7047:,
7044:b
7034:A
7025:=
7018:a
6997:,
6994:a
6984:B
6975:=
6968:A
6949:,
6946:c
6936:B
6927:=
6920:b
6899:,
6896:c
6886:B
6877:=
6870:a
6851:,
6848:c
6838:A
6829:=
6822:a
6801:,
6798:c
6788:A
6779:=
6772:b
6749:,
6746:b
6736:a
6727:=
6720:c
6677:.
6674:A
6664:c
6655:=
6645:b
6635:B
6626:=
6616:)
6613:A
6604:2
6595:(
6585:)
6582:c
6573:2
6564:(
6555:=
6545:b
6535:)
6532:B
6523:2
6514:(
6505:=
6498:a
6478:a
6458:C
6454:B
6450:c
6446:A
6442:C
6434:a
6416:B
6412:A
6408:c
6404:b
6400:a
6396:Ο
6392:C
6361:)
6358:b
6355:+
6352:a
6349:(
6343:2
6340:1
6326:)
6323:b
6317:a
6314:(
6308:2
6305:1
6290:=
6284:)
6281:B
6278:+
6275:A
6272:(
6266:2
6263:1
6249:)
6246:B
6240:A
6237:(
6231:2
6228:1
6171:c
6165:2
6162:1
6147:)
6144:B
6141:+
6138:A
6135:(
6129:2
6126:1
6112:)
6109:B
6103:A
6100:(
6094:2
6091:1
6076:=
6073:)
6070:b
6064:a
6061:(
6055:2
6052:1
6035:C
6029:2
6026:1
6011:)
6008:b
6005:+
6002:a
5999:(
5993:2
5990:1
5976:)
5973:b
5967:a
5964:(
5958:2
5955:1
5940:=
5937:)
5934:B
5928:A
5925:(
5919:2
5916:1
5900:c
5894:2
5891:1
5876:)
5873:B
5870:+
5867:A
5864:(
5858:2
5855:1
5841:)
5838:B
5832:A
5829:(
5823:2
5820:1
5805:=
5802:)
5799:b
5796:+
5793:a
5790:(
5784:2
5781:1
5764:C
5758:2
5755:1
5740:)
5737:b
5734:+
5731:a
5728:(
5722:2
5719:1
5705:)
5702:b
5696:a
5693:(
5687:2
5684:1
5669:=
5666:)
5663:B
5660:+
5657:A
5654:(
5648:2
5645:1
5595:c
5589:2
5586:1
5572:)
5569:b
5566:+
5563:a
5560:(
5554:2
5551:1
5536:=
5530:C
5524:2
5521:1
5507:)
5504:B
5498:A
5495:(
5489:2
5486:1
5463:c
5457:2
5454:1
5440:)
5437:b
5434:+
5431:a
5428:(
5422:2
5419:1
5404:=
5398:C
5392:2
5389:1
5375:)
5372:B
5369:+
5366:A
5363:(
5357:2
5354:1
5332:c
5326:2
5323:1
5309:)
5306:b
5300:a
5297:(
5291:2
5288:1
5273:=
5267:C
5261:2
5258:1
5244:)
5241:B
5235:A
5232:(
5226:2
5223:1
5199:c
5193:2
5190:1
5176:)
5173:b
5167:a
5164:(
5158:2
5155:1
5140:=
5134:C
5128:2
5125:1
5111:)
5108:B
5105:+
5102:A
5099:(
5093:2
5090:1
5044:,
5041:A
5032:+
5029:1
5026:=
5020:2
5017:A
5008:2
5000:2
4986:A
4972:,
4969:A
4957:1
4954:=
4948:2
4945:A
4936:2
4928:2
4895:)
4892:C
4886:S
4883:(
4874:)
4871:B
4865:S
4862:(
4851:)
4848:A
4842:S
4839:(
4830:S
4814:=
4807:a
4801:2
4798:1
4780:)
4777:a
4771:s
4768:(
4759:s
4748:)
4745:c
4739:s
4736:(
4727:)
4724:b
4718:s
4715:(
4702:=
4695:A
4689:2
4686:1
4666:C
4657:B
4646:)
4643:C
4637:S
4634:(
4625:)
4622:B
4616:S
4613:(
4600:=
4593:a
4587:2
4584:1
4566:c
4557:b
4546:)
4543:a
4537:s
4534:(
4525:s
4512:=
4505:A
4499:2
4496:1
4476:C
4467:B
4456:)
4453:A
4447:S
4444:(
4435:S
4419:=
4412:a
4406:2
4403:1
4383:c
4374:b
4363:)
4360:c
4354:s
4351:(
4342:)
4339:b
4333:s
4330:(
4317:=
4310:A
4304:2
4301:1
4269:,
4266:)
4263:C
4260:+
4257:B
4254:+
4251:A
4248:(
4245:=
4242:S
4239:2
4219:)
4216:c
4213:+
4210:b
4207:+
4204:a
4201:(
4198:=
4195:s
4192:2
4173:b
4169:a
4149:.
4146:A
4137:a
4128:C
4119:b
4110:+
4107:C
4098:b
4089:a
4080:b
4071:=
4064:b
4056:2
4048:a
4035:A
4026:a
4017:C
4008:b
3999:+
3996:)
3993:C
3984:b
3975:a
3966:+
3963:b
3954:a
3945:(
3939:b
3930:=
3920:A
3911:c
3902:b
3893:+
3890:c
3881:b
3872:=
3865:a
3844:c
3824:)
3821:b
3818:C
3815:a
3812:B
3809:(
3801:C
3791:B
3779:a
3769:b
3760:=
3753:C
3743:a
3722:)
3719:C
3716:a
3713:B
3710:c
3707:(
3699:B
3689:C
3677:a
3667:c
3658:=
3651:B
3641:a
3620:)
3617:a
3614:B
3611:c
3608:A
3605:(
3597:B
3587:A
3575:c
3565:a
3556:=
3549:B
3539:c
3518:)
3515:B
3512:c
3509:A
3506:b
3503:(
3495:A
3485:B
3473:c
3463:b
3454:=
3447:A
3437:c
3416:)
3413:c
3410:A
3407:b
3404:C
3401:(
3393:A
3383:C
3371:b
3361:c
3352:=
3345:A
3335:b
3314:)
3311:A
3308:b
3305:C
3302:a
3299:(
3290:C
3280:A
3268:b
3258:a
3249:=
3242:C
3232:b
3193:,
3188:)
3157:(
3146:)
3115:(
3101:)
3070:(
3059:)
3028:(
3019:=
3014:)
2983:(
2972:)
2941:(
2925:b
2921:B
2917:a
2913:C
2868:.
2865:c
2855:B
2845:A
2836:+
2833:B
2823:A
2811:=
2804:C
2791:,
2788:b
2778:A
2768:C
2759:+
2756:A
2746:C
2734:=
2727:B
2714:,
2711:a
2701:C
2691:B
2682:+
2679:C
2669:B
2657:=
2650:A
2629:A
2625:Ο
2620:a
2615:a
2611:Ο
2606:A
2580:B
2576:a
2572:A
2568:b
2561:B
2557:a
2553:c
2547:)
2543:β
2534:β
2525:β
2515:β
2508:z
2503:A
2499:c
2495:b
2489:)
2485:β
2476:β
2467:β
2437:c
2433:b
2429:a
2411:.
2405:c
2396:b
2387:a
2376:c
2367:b
2358:a
2349:2
2346:+
2343:c
2337:2
2326:b
2320:2
2309:a
2303:2
2292:1
2286:=
2276:a
2265:A
2246:c
2240:2
2231:b
2225:2
2213:2
2209:)
2205:c
2196:b
2184:a
2175:(
2169:)
2166:c
2158:2
2147:1
2144:(
2141:)
2138:b
2130:2
2119:1
2116:(
2110:=
2098:2
2093:)
2087:c
2078:b
2067:c
2058:b
2046:a
2034:(
2026:1
2023:=
2016:A
2008:2
1988:A
1972:A
1964:2
1953:1
1950:=
1947:A
1939:2
1898:.
1892:c
1883:b
1872:c
1863:b
1851:a
1839:=
1836:A
1810:.
1807:A
1798:c
1789:b
1780:+
1777:c
1768:b
1759:=
1756:a
1730:.
1727:b
1718:c
1709:+
1706:A
1697:b
1688:c
1679:=
1669:C
1666:O
1649:B
1646:O
1630:β
1624:β
1601:.
1598:)
1595:b
1585:,
1582:A
1573:b
1563:,
1560:A
1551:b
1542:(
1534::
1524:C
1521:O
1510:)
1507:c
1497:,
1494:0
1490:,
1487:c
1478:(
1470::
1460:B
1457:O
1446:)
1443:1
1439:,
1436:0
1432:,
1429:0
1426:(
1418::
1408:A
1405:O
1385:A
1381:x
1363:β
1356:z
1352:c
1342:β
1335:z
1329:β
1321:a
1315:β
1307:β
1298:a
1285:β
1276:β
1267:β
1212:.
1206:c
1195:C
1183:=
1177:b
1166:B
1154:=
1148:a
1137:A
1086:0
1078:2
1074:)
1070:b
1058:a
1049:(
1029:a
1023:a
997:0
991:c
988:,
985:b
982:,
979:a
949:.
946:C
937:b
928:a
919:+
916:b
907:a
898:=
891:c
878:,
875:B
866:a
857:c
848:+
845:a
836:c
827:=
820:b
807:,
804:A
795:c
786:b
777:+
774:c
765:b
756:=
749:a
706:β³
688:.
685:C
676:=
665:c
659:,
656:B
647:=
636:b
630:,
627:A
618:=
607:a
599:,
596:c
587:=
576:C
569:,
566:b
557:=
546:B
539:,
536:a
527:=
516:A
498:β³
491:β³
472:A
468:A
457:β³
443:β³
428:.
426:R
422:c
418:b
414:a
410:R
406:R
386:2
380:c
377:+
374:b
371:+
368:a
362:0
352:Ο
339:Ο
318:3
312:C
309:+
306:B
303:+
300:A
284:Ο
265:A
258:C
254:B
250:A
240:c
236:b
232:a
227:.
225:C
221:B
217:A
20:)
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