Talk:Spherical trigonometry

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this case an increasing radius. An editor objected to my inclusion of this information with the rationale that the page was written from the viewpoint of a fixed radius for the sphere (in fact, unity), so an increasing radius should not be mentioned. I am not sold on this rationale, but fortunately an alternative phrasing was given (that I have now adopted) that made the point moot. Unfortunately, another editor, for the sake of parallel construction, changed this back to a limit statement (without a variable). My recent edit keeps the parallel construction but removes the limit statement. In this setting, one can either talk about limits or about approximations, but if you phrase things in terms of limits you need to specify the variable in an article at this level, otherwise you are falling into the "math jargon" trap. Yes, it is easy for an experienced reader to figure out what "in the limit" is going to mean in this setting, but this is not the intended audience. Also, the terms, "reduces to" and "is a special case" which had been used here are inaccurate and misleading, so had to be removed. There is one more instance of the use of "in the limit" occurring in the last section. I have left this alone for the time being (assuming that only more sophisticated readers would get that far), but the same reasoning could be applied there.--

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definition). If this is the case, a, b, and c will all be 90 degrees as well, and the statement sin c = cos a cos b evaluates to 1=0 although it is implied by the given drawing of Napier's pentagon. As further proof, if the right spherical triangle in question is not quite a great triangle, a, b, and c may still be very close to 90 degrees, sin c would be close to 1, and cos a cos b would still be very close to 0. I think the statement should probably read "the cosine of the middle angle is equal to the product of the cosines of the opposite angles." This makes more sense in the case of the angles opposite c, however, I cannot verify it.

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fundamental in modern geometries and, perhaps more importantly, very accessible. Of course purely algebraic methods are fine, but Knowledge pages on the scalar triple product and on the parallelepiped clearly talk about their application to volumes. Given that the cosine rule has been accorded a formal proof, I see no harm in using the same basis to derive the sine law. Relating to the volume is a simple extension that does not affect the concision in my opinion. This relation to the volume is not unusual and given in

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that AB is the other side of the great circle (i.e. ellipse in the figure) that lies to the NW of the origin (also opposing poles should be at the same distance from the rim in the image, which they are not, and the polar axises be perpendicular to the major axises of the ellipses, which looks doubtful). Compare my figure to the lower right (note that the angles between the "equatorial planes" are not the same in both images).

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pole and run the side with length given, up the prime meridian. Knowing both angles to either end of the segment of fixed length ensures that the other 2 sides emanate with a uniquely determined trajectory, and thus will meet each other at a uniquely determined point, so ASA is valid. On the other hand, take side lengths of pi, pi/2, pi/2. One has a continuous family of non-congruent triangles with such measurements.

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than in my image, which is a contradiction. The sides of the polar triangle does not look like correct ellipses, which should be symmetrical around the origin (look at B'C'in the current image - in no way can that be part of an ellipse centered around the origin with the major axis equal to the diameter of the sphere, and neither can A'B' nor A'C').

402:) 06:44, 5 June 2010 (UTC) Okay, I created a new version of the figure in .svg format with the complements reversed, uploaded that to Commons, and changed the link (old and new figures are in the "spherical trigonometry" category). I then edited the text to give the correct corresponding formulae and added a description of Napier's mnemonic. 855: 1404:

There is a double implication between the two statements and I feel that is important to make this clear from the very start, in other words to say that are equivalent. The presence of two "by convention" hides this - no problem if one knows this, but may be a problem if he does not. Our disagreement

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The image of the polar triangle (upper right) is wrong. For all corners of the polar triangle to lie on the visible half of the sphere the origin in the image must lie inside ABC. As AB lies SE of the origin, the corner C' lies on the backside. The image with the poles as shown in the figure implies

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I have added a paragraph on alternative derivations of the fundamental formulae and at the same time I have given a reduced description of the proof added by recent editors. I hope the article will not be expanded to include more of the alternative derivations and thereby lose its concision. I would

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has been widely adopted in spherical trigonometry for centuries, and is found in most of the best and most historically important textbooks. It is not ambiguous, just differs slightly from conventions found in (some) modern sources which don’t typically need nearly so many half-angle trig functions.

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Certain degenerate triangles (those with antipodal vertices, e.g.,) make these congruences not generally valid, even if they are generically valid. ASA is the only one that actually holds: since "congruence" is upto an action of O(3) we can situate one of the vertices at a given angle at the south

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Yes, there is a SSS congruence for spherical triangles such that AAA is also true where Angle A = Angle B = Angle C = 270 Degrees (implying that each angle is 90 Degrees). The distance of each "side" in this "spherical" model would be the Circumference (measurement 360 Degrees) divided by 4. For a

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and the book by Jennings on Modern Geometry. I did not want the page to simply be a collection of identities and not offer accessible insight into some of the mathematics of spherical trigonometry. For example, there is no mention of dual triangles on the sphere that could also provide alternative

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Yes! the page definitely needs an overhaul and has several gaps that need filling. One request: L'Huillier's theorem is the only formula offered to calculate the spherical excess and this is used by some software packages to compute the area of a general spherical polygon. However a triangle is

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I'm not sure. This whole article contradicts itself in the way that it is now. In the example you gave, for example, it says the spherical excess formula calculates how much the triangle exceeds 180 degrees. But in the formula, it subtracts pi, not 180 degrees. On the other hand, geometry is often

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Also: the smaller triangle you start with, the greater polar triangle you will get (when the sides are all 90° the triangles are identical, and when one of the triangles is a dot, the other is a great circle). In the current image both the original and the polar triangles are smaller respectively

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I have recently made some changes in this section that I think need some additional explanation. The issue has to do with the comparison of the spherical laws with their planar counterparts. The phrasing "in the limit" that had been used is ambiguous without referring to the changing variable, in

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I've just reverted a bunch of reformats of trig formulas. These made the formulas less easy to understand, e.g., by using built-up fractions instead of a 1/2 multiplier for half-angle formulas and by introducing unnecessary parentheses. I recommend that such blanket changes be discussed on this

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I disagree with your assertion that the inclusion of parentheses is necessary to make the formulas "formally correct" (see my comment on Meridian arc talk page). (And I don't understand the point you are making about implied and explicit multiplications.) I can go either way on the square root

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I started out with the intention of adding Napier's rules but I have branched out to a general tidy up, re-ordering material where necessary. The section on identities will get some more added. The section on derivations will be condensed a little. Please check for typos. Please let me have any

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I am not sure if concision is necessarily a virtue. If someone seeks concision (s)he can simply stop reading beyond the statement and the section on Alternate derivations. Students (even in high school) and researchers often seek more than simply the statement of the results. Vector methods are

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There is an error with the statement "the sine of the middle angle is equal to the product of the cosines of the opposite angles." As a counter-example, consider the case of a great triangle, where all corners are 90 degreees (I presume this also counts as a right spherical triangle by Napier's

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defined as a volume of a parallelpiped although it can certainly be interpreted as such. Similarly the cyclic symmetry follows from algebraic definitions and it does not need a proof by the basis independence of volume. Furthermore I know of no problems using spherical trig in which the volume

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These were no blanket changes but corrections. As stated now (without the brackets), these formulas are formally incorrect. The parentheses are not optional here as there is no universally accepted or understood rule giving implied multiplication a higher priority than explicit multiplication,

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A fairly major limitation to the suggestion to use Mathjax rendering is that it can only be used by people who are logged into Knowledge. I've no idea how many people this leaves out, possibly it's 90% of readers. In any case, I've started lobbying for a way to expand the use of the Mathjax

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This entry needs some work before it is explanatory. It uses a lot of terms without defining them. For example, the formula sin A/sin a etc is stated without explaining what a, A, b, B etc refer to. The diagram is pretty but the letters in it do not correspond to those used in the entry.

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I like your edit as it provides a sensible approach to the alternative derivation "problem". I say problem because I have seen too many articles where some editors have felt that every alternative should be reported on. This may be alright if the subject of an article is the

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The choice to change the picture (made by Arif Zaman above) was an unfortunate one, because it completely loses the incredibly clever mnemonic developed by Napier for the right spherical triangle formulae; I will attempt to restore it. I also corrected the title above.

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sometimes badly characterized by its sides so the resulting area can be inaccurate. A better alternative for area calculations is one of Napier's analogies applied to the quadrangle bounded by a segment of a great circle, two meridians, and the equator for which (

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The image of the Polar Triangle shows references to A',B',C',A,B,C. Yet, the text formulas that seem to be associated with it include a,b,c,a',b',c' but these lower-case points are not shown on the image. It may be confusing for the easily confused such as me.

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i.e. area of nearly a whole hemisphere). If you allow "improper" angles (e.g. take any small spherical triangle and swap your idea of the interior and exterior) then the triangle can have area of nearly the entire surface of the sphere (spherical exces of almost

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they’re respectively large and small letters, as is more usual among navigators and consistent with the others in the section. Moreover, FWIW, the first is ‘upside down’ WRT the other two. Do we need to provide this formula twice in one article, anyway?

1741: 901:. I shall also move the Spherical excess to the end of the article (and include the above expression) although Spherical excess probably deserves a page of its own where its applications could be discussed. Perhaps a stub should be created? 685: 431:

The counter-example you give is in error because you forgot to use the complement of c (indicated by a bar over the c) in the formula. The sine of c_bar is the cosine of c, so it comes out as 0 = 0, and the formulae are correct.

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Since the image is a projection into R^2 and none of the angles are labelled with an "L" type symbol indicating an angle of pi/2, i think that the figure can be used as a general spherical triangle on S^2 without much problem...

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Also, the gamma label has been lost from the first illustration. It would also be nice if there were some way to accentuate the triangle in the figure, e.g. by heavier lines defining the triangle of interest, or by color.

1160: 1239:. There is absolutely no need to bring the parallelpiped into the discussion. The scalar triple product is simply a descriptive name of a scalar product in which one of the vectors is a vector product of two others: it is 1362:

The sides of proper spherical triangles are less than π and satisfy 0 < a + b + c < 2π; it may be shown that a spherical triangle is proper if and only if all the sides are less than pi . (Todhunter, Art.22,32).

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signs; the editor that used the 1/2 power presumably thought that this looked better. I tend to agree (the large square root sign is distracting); in any case, I would normally defer to the preferences of the editor.

356:. The equations as stated in the wiki article are the co-equations of this page. So either the equations need to be corrected, or the picture needed to have the complementary angles drawn. I chose to fix the picture. 351:

I corrected the image a bit more. The original figure was mismatched with the description of the equations given in the text. The figure, as it was previously drawn was similar to the one on the referred page:

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A naive reading suggests that there are SIX independent constraints on the triangle, moreover one may wonder why the sides' sum is 2*pi rather than 3*pi. What about replacing the second sentence by

455:, P.R.China. Maybe just experimenting to see if wikipedia is available. Or a Chinese censor trying to be disruptive? Who knows, this is just speculation... Let's just revert to the full version. 1423:

I did not make any assumptions on the nature of other readers. I only mention that your suggestion confuses me. Moreover, it did not make me understand what you explained in your last comment.

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opposite, respectively—as shown in the illustration immediately above. Adding all six labels to this drawing could make it look rather cluttered and impede visualization in three dimensions.—

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the image labeled "Spherical Triangle" is actually of a "right spherical triangle", as indicated by its German title: rechtkugel dreieck, literally "right-angle three-corner". --

678: 850:{\displaystyle \tan {\frac {E}{2}}={\frac {\sin {\frac {1}{2}}(\phi _{2}+\phi _{1})}{\cos {\frac {1}{2}}(\phi _{2}-\phi _{1})}}\tan {\frac {\lambda _{2}-\lambda _{1}}{2}}.} 1018:

Regarding readability, is there any particular reason to use the ^(1/2) form instead of the much easier to read square root form (which, again, made parentheses obsolete)?

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I think that the solution is to introduce the definition of a proper triangle elsewhere (not in the notation section) and list the properties of such triangle there.

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No idea why someone would want to remove large sections of this page as has been done recently. The last edit (which i reverted) was from someone at an IP in

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These are good questions--do SSS, SAS, ASA, and/or AAS imply congruence of spherical triangles? The answer deserves to be in the article. Does anyone know?

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Upgrade is proceeding. This article is obviously fairly heavy on mathematics. Only recently have I discovered the great improvement made the Preference-: -->

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That the sum of the angles is <3*pi and that the sum of the sides is <2*pi, are both easily explained by the limit case of 3 points on a great circle?

1839: 130: 897:. I shall therefore replace it with a link and brief comments. The section on Conruent triangles didn't really sit well here so I shall moved it to 1095: 1015:

I am open in regard to using a 1/2 fore-factor or using fractions instead, it is only that by using proper fractions we can avoid the parentheses.

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page credits J.H. Lambert with Girard's Theorem; this page Harriot. (At least you agree it wasn't Girard! :-) Perhaps there was a Harriot--: -->

1366:(perhaps another article of Todhunter is needed; I am not absolutely sure about the if-and-only-if statement, if I am wrong please tell me this) 262:

sphere, each Side then equals 2*Pi*r/4. For a Unit Sphere, this Equilateral Triangle has it's sides equa to 1.570796327 == 1.57. "I Think"

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therefore we should avoid this formally incorrect jargon where it causes difficulties for the readers to correctly interprete the formulas.

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The angles of proper spherical triangles are (by convention) less than π so that π < A + B + C < 3π. (Todhunter, Art.22,32).

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The sides of proper spherical triangles are (by convention) less than π so that 0 < a + b + c < 2π. (Todhunter, Art.22,32).

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Actually, I've just thought now to check the page for triangles. That whole page is in degrees. Should this page align to that? (

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performed using degrees. I would agree to your proposition, but I do still have reservations. I'll begin switching it over now.

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The figure showing a spherical triangle can be improved to display the projection of the circles as ellipses. Bo Jacoby.

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this artical seems over complicated by use of both degrees and radians. I propose to a change to make it entirely radians.

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By convention, where vertices are labelled with capital letters, the corresponding small letters refer to the

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Village pump (technical)#How can an user without a Knowledge account get the benefits of Mathjax rendering?

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mathJax. Is there a nice way of flagging an article which encourages readers to set this preference?

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on Knowledge. If you would like to participate, please visit the project page, where you can join

1736:{\displaystyle \sin {\tfrac {1}{2}}\varepsilon =\tan {\tfrac {1}{2}}c\,\tan {\tfrac {1}{2}}h_{c}} 1283: 1209: 536: 483: 89: 893:

The end of my upgrade is now in sight. The History section had been lifted almost verbatim from

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is the angular difference between two parallel small circles -- one is through the points

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of a result, but outside of that limited scenario I don't see that it has much value. --

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proofs to the spherical law of sines. Again, the book by Jennings is worth mentioning.

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on this evidently reflects a disagreement on the probable nature of the readers.

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As an aside, the sum of angles - pi = area formula should probably be included.

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uses Greek letters for angles and italics for sides, mathematician-style as at

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Your suggestion makes it confusing for me. What's wrong with the current text?

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Lambert progression? Anyway, I think the two pages should be consistent.

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No, but that whole section is confusingly worded and technically sloppy.

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It’s a bit late for this discussion, but note that notation of the form

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It seems necessary to say something about the radius in the sin law?

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radians as a sum of such internal angles (spherical excess of almost

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By "proper angle" it means angles less than a straight angle (i.e.

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I agree—avoid multiple derivations and keep the article concise.

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article too (or at the very least a summary and then a link to

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http://www.rwgrayprojects.com/rbfnotes/trig/strig/strig.html

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Is AAS a valid congruence theorem for spherical triangles??

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Is ASA a valid congruence theorem for spherical triangles??

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Is SAS a valid congruence theorem for spherical triangles??

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Is SSS a valid congruence theorem for spherical triangles??

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Sadly I am again reverting the edits of new wikipedian

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Please try to answer whether these questions are known:

1794:. I'd like to see this area formula described in this 1262:

http://mathworld.wolfram.com/SphericalTrigonometry.html

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no longer mentions Girard’s Theorem at all AFAICT, and

1712: 1689: 1665: 1125: 1106: 1098: 1068: 1060: 1657: 1610: 1586: 1560: 1536: 1504: 688: 660: 101:, a collaborative effort to improve the coverage of 1735: 1619: 1595: 1569: 1545: 1522: 1154: 1084: 849: 672: 638:adverse reactions. There is still more to come. 1184:be grateful if experienced editors can comment. 558:agrees with this article—with a citation, yet! — 1530:radians), so that a triangle could have almost 921:. Please consider adding to this discussion. 1147: 1119: 8: 47: 1727: 1711: 1688: 1664: 1656: 1609: 1585: 1559: 1535: 1503: 1146: 1145: 1124: 1118: 1117: 1105: 1097: 1067: 1059: 832: 819: 812: 794: 781: 764: 747: 734: 717: 708: 695: 687: 659: 340:I corrected the image for Neper's circle 550:I don’t know the facts of the case, but 1747:is the area of the spherical triangle, 1703: 49: 19: 1603:or sum of "internal" angles of almost 629:Edited 07:56, 21 September 2012 (UTC) 604:The spherical sine law as given under 7: 1523:{\displaystyle 0<\theta <\pi } 606:Spherical trigonometry#Islamic world 95:This article is within the scope of 38:It is of interest to the following 1244:interpretation is of relevance. 1085:{\textstyle \tan {\tfrac {1}{2}}a} 14: 1840:Mid-priority mathematics articles 614:Spherical trigonometry#Identities 115:Knowledge:WikiProject Mathematics 118:Template:WikiProject Mathematics 82: 72: 51: 20: 1293:The image of the polar triangle 1273:Limits in sine and cosine rules 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1076: 996:talk page first. 842: 804: 772: 725: 703: 584:comment added by 509:comment added by 499: 482:comment added by 286: 269:comment added by 218:comment added by 184: 172:comment added by 155: 154: 151: 150: 147: 146: 1847: 1808: 1800:Lexell's theorem 1793: 1789: 1782: 1775: 1765: 1761: 1757: 1750: 1746: 1742: 1740: 1739: 1734: 1732: 1731: 1722: 1713: 1699: 1690: 1675: 1666: 1649:Lexell's theorem 1628: 1626: 1624: 1623: 1618: 1602: 1600: 1599: 1594: 1578: 1576: 1574: 1573: 1568: 1552: 1550: 1549: 1544: 1529: 1527: 1526: 1521: 1280:Bill Cherowitzo 1206:Bill Cherowitzo 1161: 1159: 1158: 1153: 1151: 1150: 1135: 1126: 1123: 1122: 1116: 1107: 1091: 1089: 1088: 1083: 1078: 1069: 981: 974: 971: 968: 877:Appearance-: --> 856: 854: 853: 848: 843: 838: 837: 836: 824: 823: 813: 805: 803: 799: 798: 786: 785: 773: 765: 756: 752: 751: 739: 738: 726: 718: 709: 704: 696: 679: 677: 676: 671: 596: 521: 498: 476: 470:Some Work Needed 285: 263: 230: 208:Explicit Radius? 167: 123: 122: 119: 116: 113: 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1424: 1420: 1419: 1418: 1417: 1399: 1398: 1385: 1382: 1357: 1356: 1353: 1345: 1342: 1341: 1340: 1294: 1291: 1274: 1271: 1258: 1246:Peter Mercator 1233: 1232: 1231: 1230: 1186:Peter Mercator 1180: 1177: 1176: 1175: 1174: 1173: 1149: 1144: 1141: 1138: 1132: 1129: 1121: 1113: 1110: 1104: 1101: 1081: 1075: 1072: 1066: 1063: 1052: 1034: 1033: 1019: 1016: 1013: 992: 989: 988: 987: 937: 934: 917:renderer; see 914: 913: 903:Peter Mercator 880:Peter Mercator 874: 873: 859: 858: 857: 846: 841: 835: 831: 827: 822: 818: 811: 808: 802: 797: 793: 789: 784: 780: 776: 771: 768: 763: 760: 755: 750: 746: 742: 737: 733: 729: 724: 721: 716: 713: 707: 702: 699: 694: 691: 669: 666: 663: 640:Peter Mercator 634: 631: 601: 598: 586:89.239.205.120 575: 572: 571: 570: 526: 523: 471: 468: 448: 445: 419:Planet sputter 391: 390: 337: 334: 333: 332: 331: 330: 313: 312: 311: 310: 303: 302: 253: 252: 249: 246: 243: 235: 232: 220:71.230.115.205 209: 206: 165: 164:so for example 163: 159: 156: 153: 152: 149: 148: 145: 144: 133: 127: 126: 124: 107:the discussion 94: 93: 77: 65: 64: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 1852: 1841: 1838: 1836: 1833: 1832: 1830: 1823: 1822: 1818: 1814: 1810: 1807: 1801: 1797: 1787: 1780: 1773: 1769: 1728: 1724: 1717: 1714: 1708: 1705: 1700: 1694: 1691: 1685: 1682: 1679: 1676: 1670: 1667: 1661: 1658: 1650: 1642: 1638: 1635: 1632: 1614: 1611: 1590: 1587: 1564: 1561: 1540: 1537: 1517: 1514: 1511: 1508: 1505: 1497: 1494: 1493: 1492: 1491: 1487: 1483: 1478: 1475: 1472: 1469: 1463: 1455: 1451: 1447: 1443: 1442: 1441: 1440: 1439: 1435: 1431: 1428: 1425: 1422: 1421: 1416: 1412: 1408: 1403: 1402: 1401: 1400: 1397: 1393: 1389: 1386: 1383: 1380: 1379: 1378: 1377: 1373: 1369: 1364: 1360: 1354: 1351: 1350: 1349: 1343: 1339: 1335: 1331: 1326: 1325: 1324: 1323: 1319: 1315: 1306: 1299: 1292: 1290: 1289: 1285: 1281: 1272: 1270: 1268: 1263: 1256: 1255: 1251: 1247: 1242: 1238: 1229: 1225: 1221: 1217: 1216: 1215: 1211: 1207: 1203: 1198: 1197: 1196: 1195: 1191: 1187: 1178: 1172: 1169: 1166: 1142: 1139: 1136: 1130: 1127: 1111: 1108: 1102: 1099: 1079: 1073: 1070: 1064: 1061: 1053: 1051: 1047: 1043: 1038: 1037: 1036: 1035: 1032: 1028: 1024: 1020: 1017: 1014: 1010: 1009: 1008: 1007: 1003: 999: 990: 986: 983: 982: 976: 975: 963: 959: 955: 954: 953: 952: 948: 944: 935: 933: 932: 928: 924: 920: 912: 908: 904: 900: 896: 892: 891: 890: 889: 885: 881: 872: 868: 864: 860: 844: 839: 833: 829: 825: 820: 816: 809: 806: 795: 791: 787: 782: 778: 769: 766: 761: 758: 748: 744: 740: 735: 731: 722: 719: 714: 711: 705: 700: 697: 692: 689: 682: 681: 667: 664: 661: 652: 651: 650: 649: 645: 641: 633:Major tidy up 632: 630: 628: 624: 620: 615: 611: 607: 599: 597: 595: 591: 587: 583: 573: 569: 565: 561: 557: 553: 549: 548: 547: 546: 542: 538: 537:Jamesdowallen 534:Girard--: --> 532: 522: 520: 516: 512: 511:98.222.58.129 508: 500: 497: 493: 489: 485: 481: 469: 467: 466: 462: 458: 454: 446: 444: 443: 439: 435: 429: 428: 424: 420: 414: 413: 409: 405: 401: 397: 389: 385: 381: 376: 375: 374: 373: 370: 365: 364: 361: 357: 355: 349: 348: 345: 341: 329: 325: 321: 320:128.84.234.59 317: 316: 315: 314: 307: 306: 305: 304: 301: 297: 293: 289: 288: 287: 284: 280: 276: 272: 268: 259: 257: 250: 247: 244: 241: 240: 239: 233: 231: 229: 225: 221: 217: 207: 205: 203: 199: 195: 191: 185: 183: 179: 175: 174:138.251.30.10 171: 157: 142: 138: 132: 129: 128: 125: 108: 104: 100: 99: 91: 85: 80: 78: 75: 71: 70: 66: 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 1805: 1785: 1778: 1771: 1646: 1479: 1476: 1473: 1470: 1467: 1464:please check 1365: 1361: 1358: 1347: 1330:Episcophagus 1314:Episcophagus 1310: 1276: 1267:Joao_Nemmers 1257: 1240: 1234: 1201: 1182: 1023:Matthiaspaul 994: 978: 965: 957: 943:Tesseract501 939: 915: 875: 636: 619:Odysseus1479 612:, but under 603: 577: 560:Odysseus1479 528: 501: 473: 450: 434:Ted Sweetser 430: 415: 404:Ted Sweetser 396:Ted Sweetser 392: 366: 358: 350: 342: 339: 271:Bumppjohnson 265:— Preceding 260: 254: 237: 211: 189: 186: 161: 137:Mid-priority 136: 96: 62:Mid‑priority 40:WikiProjects 1446:Suppongoche 1407:Suppongoche 1368:Suppongoche 580:—Preceding 505:—Preceding 478:—Preceding 256:66.245.7.28 214:—Preceding 168:—Preceding 112:Mathematics 103:mathematics 59:Mathematics 1829:Categories 1344:suggestion 360:Arif Zaman 1631:jacobolus 1480:? thanks 1165:jacobolus 292:Duoduoduo 1817:contribs 1809:uantling 1766:and the 962:Odysseus 582:unsigned 507:unsigned 492:contribs 480:unsigned 279:contribs 267:unsigned 216:unsigned 194:Buggy793 170:unsigned 1647:On the 1468:should 453:Kowloon 139:on the 30:C-class 1743:where 1430:Gollem 1388:Gollem 574:Figure 484:Gnomon 344:Eroica 36:scale. 1202:proof 958:sides 190:EDIT: 1813:talk 1790:and 1515:< 1509:< 1486:talk 1450:talk 1434:talk 1411:talk 1392:talk 1372:talk 1334:talk 1318:talk 1284:talk 1250:talk 1224:talk 1220:cffk 1210:talk 1190:talk 1046:talk 1042:cffk 1027:talk 1002:talk 998:cffk 947:talk 927:talk 923:cffk 907:talk 884:talk 867:talk 863:cffk 644:talk 623:talk 590:talk 564:talk 541:talk 529:The 515:talk 488:talk 461:talk 457:Boud 438:talk 423:talk 408:talk 400:talk 384:talk 380:Boud 324:talk 296:talk 275:talk 224:talk 198:talk 178:talk 1706:tan 1683:tan 1659:sin 1634:(t) 1241:not 1168:(t) 1100:tan 1092:or 1062:tan 807:tan 759:cos 712:sin 690:tan 369:Don 131:Mid 1831:: 1819:) 1815:| 1783:, 1762:, 1709:⁡ 1686:⁡ 1677:ε 1662:⁡ 1627:). 1615:π 1591:π 1565:π 1541:π 1518:π 1512:θ 1488:) 1452:) 1436:) 1413:) 1394:) 1374:) 1336:) 1320:) 1286:) 1252:) 1226:) 1212:) 1192:) 1143:θ 1140:− 1137:π 1103:⁡ 1065:⁡ 1048:) 1029:) 1021:-- 1004:) 949:) 929:) 909:) 886:) 869:) 830:λ 826:− 817:λ 810:⁡ 792:ϕ 788:− 779:ϕ 762:⁡ 745:ϕ 732:ϕ 715:⁡ 693:⁡ 668:λ 662:ϕ 646:) 625:) 592:) 566:) 543:) 517:) 494:) 490:• 463:) 440:) 425:) 410:) 386:) 326:) 298:) 281:) 277:• 226:) 204:) 200:) 180:) 1811:( 1806:Q 1792:C 1788:* 1786:B 1781:* 1779:A 1774:* 1772:C 1764:B 1760:A 1755:c 1753:h 1749:c 1745:ε 1729:c 1725:h 1718:2 1715:1 1701:c 1695:2 1692:1 1680:= 1671:2 1668:1 1629:– 1612:5 1588:4 1577:, 1562:2 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