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is rather about a non-homogeneous quadratic polynomial in three variables, then such a thing comes from a homogeneous polynomial of four variables. Then there are two cases of the eigenvalues to consider, (++--) and (+++-). In the first case, the quadric is ruled by two families of lines (think the lines on a hyperbolic paraboloid or a hyperboloid of one sheet). As long as the plane you choose does not contain any line in either the family, it will cut out a conic. In the other case, reality becomes important (think of a plane that does not cut an elliptic paraboloid). In that case, you can restrict the quadric to the plane (say, writing everything in terms of x and y). That gives you a quadric in the plane, and the condition for that quadric to be a conic is that the coefficient matrix should be non-singular (and strictly semi-definite).
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defining them. As authors of good paper use a term without defining it only if it is not ambiguous, this is a good indication of the standards. I know that, when I read "quadric" in a paper, this should mean one of the five non-degenerated quadrics (six, if imaginary ellipsoid is relevant in the context). When cones and cylinders must be taken into account, authors generally use "quadratic surface". In this case, only the context allows knowing if non-irreducible quadratic algebraic sets should be included. I have tried to take this into account in my edit, but some clarification is certainly still needed. In particular the distinction between "quadric" and "quadratic surface" deserve to be mentioned.
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are nine true quadrics", and specifies these as 4 degenerate quadrics and 5 non-degenerate ones, "which are detailed in the following table". Then a paragraph says "there are 17 such normal forms. Of these 16 forms, ...". Then a table is given, consisting of a non-degenerate section containing 10 and a degenerate section containing 6.
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Regarding
Question 2, with "This pametrization establishes a bijection between a projective conic section and the projective line", might we instead say more generally that "This pametrization establishes a bijection between a quadric and a projective space"? Also, what should we do about the caveat
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starts out by giving some equations and permissible epsilon values, for a total of 17 possibilities. Then it mentions "these 17 normal forms", and mentions 7 of them that are imaginary only, single-real-point, or otherwise trivial. That leaves 10 remaining. Then the next paragraph begins "Thus, there
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quadratic in three-dimensions? The coefficient matrix is then 3×3, and the condition is that it be a non-singular (strictly) semi-definite matrix (that is, with eigenvalues of both positive and negative sign). Then any plane that avoids the origin cuts that cone in a conic section. If the question
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is satisfied by every triangle; it leads to a
Heronian triangles if and only if each of the variables is positive and rational. Yes, there is a way to frame Heronian triangles as using a quartic equation, but that doesn't make the quadratic approach any less valuable. Yes, there are other ways to
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I couldn't find this in the article: Given a quadric equation in three dimensions, what are the necessary and sufficient conditions on the parameters or parameter matrix for it to describe a conic in a two-dimensional flat of R^3? And what further conditions on the matrix characterize the different
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This is a case where terminology is not uniformly fixed. IMO, there are two methods to find the dominant one. The first is is boring, and consists in consult many textbooks for extracting the dominant terminology. The second one is to look what is meant in the articles that use the terms without
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The
English definition of a quadric NOW does not comply with all (?) other WIKIs. And there is a contradiction: The definition excludes pairs of planes, the list of quadrics below contains such objects. There are two major subclasses of quadrics: nondegenerate and degenerate. Both are quadrics.
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Goldenrod is less difficult to see, but still difficult. Actually I don’t think the image needs a color-coordinated caption. If the colors are completely removed from the caption, the specific eccentricities clarify the correspondence. I suggest we make the caption text entirely
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then an editor can request that there be a citation. Go ahead and remove it if you wish. It's been a while since I've had my algebraic varieties class, so odds are long that I will find a
Knowledge-quality citation, but I'll keep my eyes open. (Yes, we could clarify that
965:). I have also replaced the last paragraph before the table by a summary of the classification. In fact, this paragraph classified implicitly the imaginary ellipsoid among the degenerate quadrics, which contradicts the definition of "degenerate" given in
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In the current version of the article I read the sentence: "This define affine quadrics." Either grammatically or spelling-wise there seems to be an error in this sentence, which obscures its meaning to me. What did the author intend to write?
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Point 2:Your second question is the answer to the first one: in general, the parametrization does not cover the whole quadric; that is, the points of the intersection of the quadric and its tangent hyperplane at
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About the table: If one omits the "special cases", the table gives 5 non-degenerate and 3 degenerate quadrics. It could be less confusing to move the special cases in one or two specific sections of the table.
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My calculus textbook indicates that such a set is degenerate: "In three-space . . . any second-degree equation which does not reduce to a cylinder, plane, line, or point corresponds to a surface which we call
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758:. Quadric surfaces are classified into six types, and it can be shown that every second-degree equation which does not degenerate into a cylinder, a plane, etc., corresponds to one of these six types." —
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Do you have literature that refers to this as "rational parametrization"? Given that it is a mapping from (most of) the quadric to a projective space, the natural name for it (to me) is a "projection".
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The first interpretation is the correct one. However, as "This" is ambiguous, I have moved the definition at the beginning of the section, and added explanations for clarifying the end of the section.
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are not parametrized. However, I have fixed an error: not every point of the projective space ot the parameters gives a point of the quadric: the directions contained in the tangent hyperplane at
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I hope that someone familiar with these can clarify how these different passages relate to each other numerically, and which ones in one passage coincide with which ones in each other passage.
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Likewise since z_0^2 + ... + z_n^2 is a homogeneous polynomial, the standard quadric may alternatively be seen as a complex projective hypersurface in complex projective (n+1)-space CP^(n+1).
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1400:, cases b and c are not both called "tangent" but they are in this newly rewritten section. Let's say something about this difference of usage, or otherwise make things more consistent.
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The section is about rational points of quadrics, and the primitive
Heronian triangles are in one to one correspondence with the rational points of the quartic obtained by squaring
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The assertions contained in the paragraph are true, but not evident (I had to think about a while for finding a proof). So they must be sourced, proved, linked, or easy to find in
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This is a bijection between the points of the projective conic section and the points of the projective line. The fact that the points of projective spaces are often defined as
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attesting that "standard quadric" is a common name forthis concept (Looking on
Scholar Google, I did not find any source using "standard quadric" in this sense)
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The standard quadric in complex (n+1)-space C^(n+1) is defined as the set of points (z_0, ..., z_n) in C^(n+1) such that z_0^2 + ... + z_n^2 = 0.
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I'll remove this paragraph because of the following issues, which, all together, make that this paragraph does not belong to this section.
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for a triangle with those half-angle tangents. This triangle can be scaled to a similar triangle whose side lengths and area are integers.
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I'm not 100% sure that the recent efforts qualify as an improvement yet, but thank you for the work ... let's keep at it! Some questions:
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thank you for your replies and article edits. Regarding
Question 1, elsewhere in the article we number the non-projective dimensions as
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Thus rational parametrization can be used to generate all such triples of rational tangents. These in turn give rational side lengths
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for consistency with the rest of the article? I apologize for not quite following this well enough that I can answer my own question.
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Is a conic section a quadric ? In German text-books there are pairs of lines (planes in space) quadrics. Is there a cultural gap ?--
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contradicts the beginning of the section. So, this is this section that must be edited. Neverttheless I have added a clarification.
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1519:"Rational parametrization" is a standard terminology that is used in several Knowledge articles. So, I have created the redirect
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is a triangle whose side lengths and area are integers. The interior angles of a
Heronian triangle have half-angle tangents,
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Howeverm I have added some clarification. Normally, this clarification should belong to another article, such as
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As far as I know, the main property of what you call "standard quadric" is already in the article, stated as
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I am skeptical. What are the conic section and affine transformation that produces a hyperbolic parabaloid?
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Parametrizing this quadric does not requires the method described in this article, as the parametrization
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that has been equated to 1, since otherwise one would not have an indexing by the first positive integers.
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The colors in the image in the
Euclidean Plane section are mismatched from the description of the image.
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In complex projective space all of the nondegenerate quadrics become indistinguishable from each other
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on 2 October 2017. For the contribution history and old versions of the redirected page, please see
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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complex projective transformations, the standard quadrics are exactly the nondegenerate quadrics
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Is "projective conic section" the same as "quadric"? If so, let's consistently use the latter.
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are contained in the quadric. More generally, for every point of a hyperbolic paraboloid or a
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This is an important construction in mathematics and should not be omitted from the article.
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Thank you, this is coming along nicely. Another question (to demonstrate my ignorance): if
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I have added examples as a new subsection. This should be a clarification for many readers.
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for points at infinity. I admit to confusion for the present case ... is the use of index
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Can you reliably source your assertion? A far as I know, for most authors, a quadric is an
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are the coordinates of a direction vector in the space that contains the quadric. So the
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for the additional dimension used in projective geometry, whose value is typically
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Notable properties of these hypersurfaces that deserve to be added in this article
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A quadric may have any dimension. A conic section is a quadric of dimension one.
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I'm not clear on what exactly you are asking. Do you want the conditions for a
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solve this quadratic, but there are other ways to solve the circle too; e.g.,
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after having edited this article. I have linked the section to this redirect.
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1131:. If there is no objection, I shall remove the tag on "multiple issues".--
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Several rational parametrizations of the "Heronian quartic" are given in
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is a non-singular point of the quadric, is it impossible that the line
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671:{\displaystyle {\begin{bmatrix}Q&R/2\\R^{2}/2&S\end{bmatrix}}}
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1366:" -- Each point of the quadric (not in the tangent hyperplane of
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I hope someone adds information about the standard quadric
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that are rational and satisfy the quadratic relationship
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must be indexed in the same way as the space coordinates
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that the points in the tangent hyperplane are excluded? —
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always gives something that is tangent to the quadric at
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for points not at infinity, but why is it okay to assume
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which redirects to the first occurence of the phrase in
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As the directions form a projective space of dimension
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So this way of parametrizing heronian triangles seems
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has recently added the following paragraph in section
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Revision of section "Projective quadrics over fields"
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Protter, Murray H.; Morrey, Jr., Charles B. (1970),
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It is fixed now. Thank you for pointing this out.—
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259:Hyperbolic Parabaloid and Affine Transformations
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1582:for points not at infinity and is
1398:Quadric § Intersection with a line
958:{\displaystyle \varepsilon _{3}=0}
708:a quadric ! (s. recent change). --
549:{\displaystyle x^{T}Qx+x^{T}R+S=0}
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3197:Low-priority mathematics articles
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809:Vorlesungen über höhere Geometrie
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283:Conic section embedded in R^3
170:Quadric (projective geometry)
109:and see a list of open tasks.
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807:For example: Oswald Giering
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485:in scalars, or if you prefer
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1127:I tried to improve section
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1947:{\displaystyle A=(0,0,0).}
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1514:§ Intersection with a line
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1165:define affine quadrics."?
2911:I agree that if it isn't
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1078:Help:Displaying a formula
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1199:Rational Parametrization
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967:Degeneracy (mathematics)
905:The numbers don't add up
811:, Springer-Verlag,p. 61.
141:project's priority scale
1506:has nothing to to here.
1332:{\displaystyle T_{n}=0}
1299:{\displaystyle T_{n}=0}
1266:{\displaystyle t_{n}=1}
1233:{\displaystyle t_{0}=1}
1146:Grammar and/or Spelling
911:Quadric#Euclidean space
98:WikiProject Mathematics
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678:must have rank one.
673:
551:
470:
167:The contents of the
3045:
3010:
2920:
2804:
2706:
2614:
2445:
2441:and a rational area
2386:
2322:
2259:
2170:
2140:
2110:
2081:
1996:
1958:
1911:
1900:{\displaystyle z=xy}
1882:
1775:
1748:
1737:{\displaystyle n-1,}
1719:
1689:
1662:
1635:
1464:
1437:
1370:
1343:
1310:
1277:
1244:
1211:
1050:
991:Great job – thanks!
936:
885:(2nd ed.), Reading:
606:
499:
342:
121:mathematics articles
3163:. This means that,
2563:
2536:
2509:
2427:
2363:
2300:
1535:Parametric equation
1531:Parametric equation
1489:Parametric equation
1158:affine quadrics."?
1154:Was it "This define
3084:
3032:
2995:
2879:
2783:
2689:
2570:
2549:
2522:
2495:
2431:
2413:
2372:
2368:
2349:
2309:
2305:
2286:
2245:
2156:
2126:
2097:
2058:Heronian triangles
2020:
1982:
1944:
1897:
1788:
1761:
1734:
1705:
1675:
1648:
1477:
1450:
1376:
1349:
1329:
1296:
1263:
1230:
1066:
1065:
955:
668:
662:
546:
465:
90:Mathematics portal
34:content assessment
3082:
2798:Heronian triangle
2781:
2568:
2429:
2365:
2302:
2075:Heronian triangle
2067:§ Rational points
1379:{\displaystyle A}
1352:{\displaystyle A}
1168:Or anything else?
733:algebraic variety
725:algebraic variety
288:types of conics?
256:
255:
195:
194:
155:
154:
151:
150:
147:
146:
3204:
3100:
3093:
3091:
3090:
3085:
3083:
3081:
3080:
3065:
3057:
3056:
3041:
3039:
3038:
3033:
3022:
3021:
3004:
3002:
3001:
2996:
2988:
2987:
2978:
2977:
2965:
2964:
2955:
2954:
2942:
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2932:
2931:
2888:
2886:
2885:
2880:
2872:
2871:
2862:
2861:
2849:
2848:
2839:
2838:
2826:
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2816:
2815:
2792:
2790:
2789:
2784:
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2769:
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2756:
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2737:
2736:
2718:
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2698:
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2690:
2682:
2681:
2672:
2671:
2659:
2658:
2649:
2648:
2636:
2635:
2626:
2625:
2605:WP:Verifiability
2579:
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2576:
2571:
2569:
2567:
2562:
2557:
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2530:
2508:
2503:
2484:
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2326:
2318:
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2301:
2299:
2294:
2278:
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2263:
2254:
2252:
2251:
2246:
2238:
2237:
2228:
2227:
2215:
2214:
2205:
2204:
2192:
2191:
2182:
2181:
2165:
2163:
2162:
2157:
2152:
2151:
2135:
2133:
2132:
2127:
2122:
2121:
2106:
2104:
2103:
2098:
2093:
2092:
2029:
2027:
2026:
2021:
1991:
1989:
1988:
1983:
1953:
1951:
1950:
1945:
1906:
1904:
1903:
1898:
1858:
1851:
1841:
1809:
1805:
1797:
1795:
1794:
1789:
1787:
1786:
1770:
1768:
1767:
1762:
1760:
1759:
1743:
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1735:
1714:
1712:
1711:
1706:
1701:
1700:
1684:
1682:
1681:
1676:
1674:
1673:
1657:
1655:
1654:
1649:
1647:
1646:
1614:
1597:
1593:
1589:
1585:
1581:
1577:
1573:
1569:
1565:
1527:Rational variety
1524:
1493:Rational variety
1486:
1484:
1483:
1478:
1476:
1475:
1459:
1457:
1456:
1451:
1449:
1448:
1413:
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1377:
1358:
1356:
1355:
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1338:
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1322:
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1305:
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1297:
1289:
1288:
1272:
1270:
1269:
1264:
1256:
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1239:
1237:
1236:
1231:
1223:
1222:
1075:
1073:
1072:
1067:
1064:
964:
962:
961:
956:
948:
947:
897:
896:
878:
682:
677:
675:
674:
669:
667:
666:
651:
646:
645:
629:
555:
553:
552:
547:
530:
529:
511:
510:
474:
472:
471:
466:
389:
388:
373:
372:
357:
356:
312:
251:
235:
207:
199:
186:
164:
163:
157:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
3212:
3211:
3207:
3206:
3205:
3203:
3202:
3201:
3182:
3181:
3149:reliable source
3121:
3096:
3072:
3048:
3043:
3042:
3013:
3008:
3007:
2979:
2969:
2956:
2946:
2933:
2923:
2918:
2917:
2863:
2853:
2840:
2830:
2817:
2807:
2802:
2801:
2770:
2762:
2747:
2728:
2709:
2704:
2703:
2673:
2663:
2650:
2640:
2627:
2617:
2612:
2611:
2594:rational points
2590:Heron's formula
2485:
2474:
2464:
2454:
2450:
2443:
2442:
2406:
2395:
2391:
2384:
2383:
2342:
2331:
2327:
2320:
2319:
2279:
2268:
2264:
2257:
2256:
2229:
2219:
2206:
2196:
2183:
2173:
2168:
2167:
2143:
2138:
2137:
2113:
2108:
2107:
2084:
2079:
2078:
2060:
1994:
1993:
1956:
1955:
1909:
1908:
1880:
1879:
1854:
1843:
1839:
1807:
1803:
1778:
1773:
1772:
1751:
1746:
1745:
1717:
1716:
1692:
1687:
1686:
1665:
1660:
1659:
1638:
1633:
1632:
1610:
1595:
1591:
1587:
1583:
1579:
1575:
1571:
1567:
1559:
1520:
1467:
1462:
1461:
1440:
1435:
1434:
1409:
1368:
1367:
1341:
1340:
1313:
1308:
1307:
1280:
1275:
1274:
1247:
1242:
1241:
1214:
1209:
1208:
1201:
1148:
1125:
1048:
1047:
1011:
939:
934:
933:
907:
902:
901:
900:
880:
879:
875:
731:that is not an
702:
680:
661:
660:
655:
637:
634:
633:
620:
610:
604:
603:
566:for 3×1 vector
521:
502:
497:
496:
380:
364:
348:
340:
339:
310:
285:
261:
247:
236:
230:
212:
182:
161:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
3210:
3208:
3200:
3199:
3194:
3184:
3183:
3180:
3179:
3157:
3156:
3155:
3152:
3120:
3117:
3116:
3115:
3079:
3075:
3071:
3068:
3063:
3060:
3055:
3051:
3031:
3028:
3025:
3020:
3016:
2994:
2991:
2986:
2982:
2976:
2972:
2968:
2963:
2959:
2953:
2949:
2945:
2940:
2936:
2930:
2926:
2895:
2894:
2878:
2875:
2870:
2866:
2860:
2856:
2852:
2847:
2843:
2837:
2833:
2829:
2824:
2820:
2814:
2810:
2794:
2779:
2776:
2773:
2768:
2765:
2759:
2754:
2750:
2746:
2743:
2740:
2735:
2731:
2727:
2724:
2721:
2716:
2712:
2700:
2688:
2685:
2680:
2676:
2670:
2666:
2662:
2657:
2653:
2647:
2643:
2639:
2634:
2630:
2624:
2620:
2608:
2601:Heron triangle
2597:
2566:
2561:
2556:
2552:
2548:
2545:
2542:
2539:
2534:
2529:
2525:
2521:
2518:
2515:
2512:
2507:
2502:
2498:
2494:
2491:
2488:
2481:
2477:
2471:
2467:
2461:
2457:
2453:
2425:
2420:
2416:
2412:
2409:
2402:
2398:
2394:
2371:
2361:
2356:
2352:
2348:
2345:
2338:
2334:
2330:
2308:
2298:
2293:
2289:
2285:
2282:
2275:
2271:
2267:
2244:
2241:
2236:
2232:
2226:
2222:
2218:
2213:
2209:
2203:
2199:
2195:
2190:
2186:
2180:
2176:
2155:
2150:
2146:
2125:
2120:
2116:
2096:
2091:
2087:
2059:
2056:
2055:
2054:
2053:
2052:
2051:
2050:
2049:
2048:
2047:
2046:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1954:the two lines
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1896:
1893:
1890:
1887:
1822:
1799:
1785:
1781:
1758:
1754:
1733:
1730:
1727:
1724:
1704:
1699:
1695:
1672:
1668:
1645:
1641:
1602:
1601:
1600:
1599:
1554:
1553:
1540:
1539:
1538:
1517:
1510:
1507:
1500:
1474:
1470:
1447:
1443:
1405:
1404:
1401:
1394:
1391:
1375:
1360:
1348:
1328:
1325:
1320:
1316:
1295:
1292:
1287:
1283:
1262:
1259:
1254:
1250:
1229:
1226:
1221:
1217:
1200:
1197:
1196:
1195:
1147:
1144:
1124:
1121:
1120:
1119:
1118:
1117:
1116:
1115:
1095:
1094:
1093:
1092:
1063:
1060:
1057:
1041:
1040:
1010:
1009:Color Mismatch
1007:
1006:
1005:
1004:
1003:
986:
985:
970:
954:
951:
946:
942:
906:
903:
899:
898:
887:Addison-Wesley
872:
871:
867:
866:
865:
864:
863:
862:
861:
860:
859:
858:
857:
833:
832:
831:
830:
829:
828:
827:
826:
812:
798:
797:
796:
795:
794:
793:
773:
772:
771:
770:
748:
747:
701:
698:
697:
696:
695:
694:
693:
692:
681:Sławomir Biały
665:
659:
656:
654:
650:
644:
640:
636:
635:
632:
628:
624:
621:
619:
616:
615:
613:
595:
594:
593:
592:
574:
573:
572:
571:
561:
560:
559:
558:
557:
556:
545:
542:
539:
536:
533:
528:
524:
520:
517:
514:
509:
505:
489:
488:
487:
486:
480:
479:
478:
477:
476:
475:
464:
461:
458:
455:
452:
449:
446:
443:
440:
437:
434:
431:
428:
425:
422:
419:
416:
413:
410:
407:
404:
401:
398:
395:
392:
387:
383:
379:
376:
371:
367:
363:
360:
355:
351:
347:
332:
331:
330:
329:
323:
322:
311:Sławomir Biały
284:
281:
270:192.249.47.195
260:
257:
254:
253:
241:
238:
237:
232:
228:
226:
223:
222:
214:
213:
208:
202:
193:
192:
165:
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:
3209:
3198:
3195:
3193:
3190:
3189:
3187:
3178:
3174:
3170:
3166:
3162:
3158:
3153:
3150:
3146:
3145:
3143:
3142:
3141:
3140:
3136:
3132:
3127:
3124:
3118:
3114:
3110:
3106:
3102:
3099:
3077:
3073:
3069:
3066:
3061:
3058:
3053:
3049:
3029:
3026:
3023:
3018:
3014:
2992:
2989:
2984:
2980:
2974:
2970:
2966:
2961:
2957:
2951:
2947:
2943:
2938:
2934:
2928:
2924:
2914:
2910:
2909:
2908:
2907:
2903:
2899:
2892:
2876:
2873:
2868:
2864:
2858:
2854:
2850:
2845:
2841:
2835:
2831:
2827:
2822:
2818:
2812:
2808:
2799:
2795:
2777:
2774:
2771:
2766:
2763:
2757:
2752:
2748:
2744:
2741:
2738:
2733:
2729:
2725:
2722:
2719:
2714:
2710:
2701:
2686:
2683:
2678:
2674:
2668:
2664:
2660:
2655:
2651:
2645:
2641:
2637:
2632:
2628:
2622:
2618:
2609:
2606:
2602:
2598:
2595:
2591:
2587:
2586:
2585:
2581:
2559:
2554:
2550:
2546:
2543:
2532:
2527:
2523:
2519:
2516:
2505:
2500:
2496:
2492:
2489:
2479:
2475:
2469:
2465:
2459:
2455:
2451:
2423:
2418:
2414:
2410:
2407:
2400:
2396:
2392:
2369:
2359:
2354:
2350:
2346:
2343:
2336:
2332:
2328:
2306:
2296:
2291:
2287:
2283:
2280:
2273:
2269:
2265:
2242:
2239:
2234:
2230:
2224:
2220:
2216:
2211:
2207:
2201:
2197:
2193:
2188:
2184:
2178:
2174:
2153:
2148:
2144:
2123:
2118:
2114:
2094:
2089:
2085:
2076:
2070:
2068:
2064:
2057:
2045:
2041:
2037:
2033:
2014:
2011:
2008:
2005:
2002:
1976:
1973:
1970:
1967:
1964:
1941:
1935:
1932:
1929:
1926:
1923:
1917:
1914:
1894:
1891:
1888:
1885:
1878:
1875:Consider the
1874:
1873:
1872:
1868:
1864:
1860:
1857:
1850:
1846:
1837:
1836:
1835:
1831:
1827:
1823:
1821:
1817:
1813:
1800:
1783:
1779:
1756:
1752:
1731:
1728:
1725:
1722:
1702:
1697:
1693:
1670:
1666:
1643:
1639:
1631:Point 1: the
1630:
1629:
1628:
1624:
1620:
1616:
1613:
1606:
1605:
1604:
1603:
1563:
1558:
1557:
1556:
1555:
1552:
1548:
1544:
1541:
1536:
1532:
1528:
1523:
1518:
1515:
1511:
1508:
1505:
1501:
1498:
1494:
1490:
1472:
1468:
1445:
1441:
1432:
1431:
1430:
1429:
1428:
1427:
1423:
1419:
1415:
1412:
1402:
1399:
1395:
1392:
1389:
1373:
1365:
1361:
1346:
1326:
1323:
1318:
1314:
1293:
1290:
1285:
1281:
1260:
1257:
1252:
1248:
1227:
1224:
1219:
1215:
1206:
1205:
1204:
1198:
1194:
1190:
1186:
1182:
1181:
1180:
1179:
1175:
1171:
1166:
1164:
1161:Or was it "Th
1159:
1157:
1152:
1145:
1143:
1142:
1138:
1134:
1130:
1122:
1114:
1110:
1106:
1101:
1100:
1099:
1098:
1097:
1096:
1091:
1087:
1083:
1079:
1061:
1058:
1055:
1045:
1044:
1043:
1042:
1039:
1035:
1031:
1027:
1026:
1025:
1024:
1020:
1016:
1008:
1002:
998:
994:
990:
989:
988:
987:
984:
980:
976:
971:
968:
952:
949:
944:
940:
931:
930:
929:
928:
924:
920:
915:
912:
895:
892:
888:
884:
877:
874:
870:
856:
852:
848:
843:
842:
841:
840:
839:
838:
837:
836:
835:
834:
825:
821:
817:
813:
810:
806:
805:
804:
803:
802:
801:
800:
799:
792:
788:
784:
779:
778:
777:
776:
775:
774:
769:
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184:its history
112:Mathematics
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30:Start-class
3186:Categories
869:References
173:page were
2063:Quantling
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3169:D.Lazard
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579:Thanks,
244:365 days
210:Archives
2913:WP:CALC
1460:and no
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