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assumption for humans to have made until experimental evidence contradicted it. No revision of mathematics was necessary. We merely accept that mathematical things don't work like real things, and keep going. We fill up
Euclidean planes (which have zero thickness) with lines that have zero thickness, which contain points of zero anything. A sphere's surface has zero thickness. Mathematics is FULL of things that can't exist in the physical world. Why should "physical impossibility" be permitted as a discussion-point for this horn when it's not a discussion-point for zero-thickness things that are all over the place in mathematics? Either the area under the curve is finite or it isn't. If it's finite, then it's perfectly reasonable to talk about painting this horn's interior using a finite amount of paint.
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with the finite amount of paint that we know fills it. Insert the smaller horn all the way into the interior of the larger horn. (It's even possible to calculate the finite volume of paint that will be forced to overflow the flared rim of the larger horn by the displacement.) Let it sit for awhile. Remove the smaller horn from within the larger horn. It will come out having been painted by having been dipped. This method is obvious but I'm just not finding it anywhere. Please do the research and add it in.
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obviously a bit of a cheat. However, if you're going to run with the paint thing, then the infinite time needed to complete the coverage does become important, because it's actually the paradox reasserting itself in a restated way - the reason the time required is infinite is simply that the length (thus area) of the horn is infinite, and there's no getting around that. -
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2217:, then the resulting volume is also finite and looks like a glass tumbler or jar infinitely extended in one direction. Meanwhile the volume inside the rotated cissoid is clearly infinite, and it is the comparison of these two volumes which is interesting. The area here is not a surface area as in Gabriel's Horn, but in effect a cross-sectional area.--
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I'm sure this is a fascinating subject. Do you suppose this could be rewritten so the average layperson (say, with a US high school education) could have the foggiest idea of what it means? I'm trying to understand how an object with finite surface area could have infinite volume, and unfortunately
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I am a novice. However, one thing strikes me. PI being transcedental we can never find out how much paint is needed to fill it(otherwords we do not know the volume but we have a name for what we do not know: PI). Therefore infinite area might suggest absurdity and not unpaintable infinate surface. We
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I do not understand the infinite time or "length (hence area)" points. We know that "length infinite, so volume infinite" would be wrong. We could come up with a another horn based on y=1/x^2 where both the area and the volume would be finite, but the length and time would still be infinite; somehow
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It isn't known as that in most places; in the 17th to 19th century everyone just called it the "solidum hyperbolicum acutum" ("solido hyperbolico acuto" being the ablative case, of course) or translations of the same into
Italian/French/English/German/whatever. We've got a published source that has
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At the end of
Mathematical Definition it says "...it was considered paradoxical as, by rotating an infinite _area_ about the x-axis, an object of finite volume is obtained.". I changed it to read 'curve' instead of 'area', because building a solid of revolution is rotating a curve, not an area, the
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No offense intended, but you've got to be kidding. "What in particular is the problem?"? Your response indicates to me that you are probably either a professional or enthusiastic amateur in a field that uses advanced mathematics on a regular basis. Unfortunately, it's all too easy when one is an
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More realistically though, I'm gonna say "no" because any substance sufficiently thin to have a chance of "painting" the horn (it would need to be infinitely thin) would cease to have any properties one would associate with a liquid or even matter, let alone "paint" and therefore you couldn't even
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c1c2c3c4c- it is not a paradox, just flawed logic. as y approaches x it never equals o it is always .00000001, .00000000001 etc thus volume is created infinitely. even after fixing that mathematical mistake, infinity is a physical impossibility the entire idea of this being a paradox even assuming
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I have tried to find a source to document one method for painting the EXTERIOR of this horn. Simply make a horn that was built spinning the graph "y=2/x". Every point on this curve is twice as far from the x-axis as the same point on the "y=1/x" curve having the same x-value. Fill the larger horn
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The paint analogy was once in the article and while its a cute way of illustrating the paradox, it simply offered too many problems and was ultimately edited out. See above discussion at top of this talk page on the subject. But to answer your question, "If you filled the horn with paint and then
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would be clearer than the calculations on the main page. Please note that I did specify I was looking for an article that the average layperson (say, with a US high school education) could follow. I don't believe most high school students graduate with an understanding of calculus sufficient to
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What in particular is the problem? The horn extends an infinite length to the right. So measuring from the left to a particular point, its surface area and volume both increase as the point moves to the right. Because of its particular shape, the surface area increases without limit, while the
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It seems an odd method of explanation to me also. Since we seem to be in agreement, I have removed it and added a resolution to the paradox that hopefully explains things in a more clear-cut way (I think that the best way to illustrate this would be with some diagrams; hopefully someone will come
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Unfortunately the paint conceptualisation, while clever, misses the essence of the paradox. The problem rests on the apparent incompatibility of the formulas of area and volume where infinite quantities are concerned, at least in some circumstances. Coming up with an infinitely thinnable paint is
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The final sentence reads "If the paint is considered without thickness, it would further take infinitely long time for the paint to run all the way down to the "end" of the horn." To me it seems irrelevant how much time it would take to paint the horn (since I'm not paying anyone to do it ;). The
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I don't think we should be discussing the paint using concepts like molecules and friction, etc. Those come from the real world, not mathematics. What did Euclid know about molecules? The idea that for any quantity of water "x" there is a quantity of water that is "x/2" is a perfectly reasonable
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has "Hyperbolicum Acutum", for example. "acute hyperbolic solid" is even given in Julian Havil's controversial-to-this-talk-page book, on page 83. (I hope that this conotroversy is addressed now, by the way, by the mathematics professor that not only made the same error as Havil, but documented
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And of course not only couldn't you paint it in theory, but also not in practice, regardless if the horn had a truely infinite area or not. Any substance that one could qualify as being "paint", would be far to thick to reach the tiny recess of the narrow part of the horn. Indeed, the molecules
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The paint thing is taken off, but you CANNOT fill it with infinitely thin paint. (as if you could even see it). It would take FOREVER for you to pour paint into it, thus you could pour 20 times the amount it should hold, and it STILL wouldn't fill. Assuming the end of the horn itself generates
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Well, it seems as though in the context of the horn, the paradox mentioned defeats itself. A finite volume of paint could have an infinite surface area, if a piece of metal can also have those two quantities... Either I am missing something, or the original creators of the paradox were missing
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Third, and here I'm doubtless going to display my ignorance, this appears to my untrained eye to be one of those cases of subsets of infinities. The diameter of the "horn" decreases but since the horn extends to infinity then it would seem that the volume it holds is also
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Gabriel's horn has an infinite surface area, so in theory you could not paint one. But since
Gabriel's Horn has a volume of π cubic units, one could supposedly fill it with π cubic units of paint. If you did this, and then emptied the horn, would it be painted?
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I have added what I hope is a reasonable explanation. It could do with some work, but I made an effort to try and explain it without much reference to the mathematics involved; it's quite a difficult thing to understand without it, however. The diagrams on
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rotated around the y axis and claims the same thing (infinite volume enclosed by a finite-area surface) without giving a proof. The author cites a portion of a letter from Sluse to
Huygens, dated 12 April 1658. This letter can be found in
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As far as I can determine, only authors of recreational mathematics books, and the odd textbook, use the name "Gabriel's horn". Even that is a late 20th and 21st century affectation; there being nothing before the 1980s that I can find.
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something (I shall not say what). Does anyone have any external information about the "paradox"? The time argument seems to be of a completely different paradox. btw, I am changing nothing as this is my own personal opinion... --
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rotated about the x-axis, its volume can be estimated by dividing the rotation into slices and adding the sum of their volumes. By making the slices infinitely thin, their volume approaches the definite integral of
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It is somewhat saddening that the two names first introduced for this are not the thing's proper name, promoting a false notion of what its primary name is, and a supposed connection to
Christianity that isn't even
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I wondered the same thing the first time reading this article. Back then it actually had the paint analogy contained in the article. Thinking strickly in theory on your question and it can really have you in
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follow the math on the main page (whether that's a crying shame or not is a separate subject). I know I didn't, and my challenge is compounded by the fact that I graduated college over 25 years ago.
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I suppose I'm simply going to have to accept that this has been figured out by better brains than mine. It is counterintuitive that a shape of infinite length could have less than infinite volume.
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The article could be improved by adding a bit of history to it. Perhaps a whole section with it. When and where did
Evangelista Torricelli first describe Grabriel's Horn? And the like. --
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researched the names into the 18th century, and I can confirm from my own findings that mathematics dictionaries into the 19th century continued likewise. Barlow's 1814
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themselves would be too large. Conversely, consider filling the finite volume and shape of the horn. It would be very easy to fill such a shape with paint even if it
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The title and most of the external links spell "Horn" capitalized. The article lowercases it as "horn." I think we should settle on one spelling. What should it be?
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area is a mere consequence of the rotating curve. If anyone disagrees I'd like to hear their opinion, please. Christophe
Lasserre 16:23, 16 November 2007 (UTC)
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approaches infinity yields infinity itself. Therefore, the paradox forms that an infinite sum of infinities equals a finite number - in this case,
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making that error.) The original Latin "solidum hyperbolicum acutum" can be found even in this century in non-recreational mathematics books;
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As far as I can tell the point with the
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volume does not exceed a particular number. You can show this by doing the calculations. If you find this helpful, put it in the article. --
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A limit was evaluated as being equal to infinity which is nonsensical; I rephrased it for correctness. Jake 06:52, 27 November 2006 (UTC)
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Another interesting paradox of
Gabriel's Horn is the "Infinite Sums of Infinity" paradox. Since Gabriel's horn is the function of
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relevant point is that if the paint is considered to be without thickness, then any volume of paint can cover any surface area.
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It all depends how plain you want it. As you move right, you add a smaller amount each time. Consider the following series:
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is wrong. If the cup has infinite height, it must have infinite area, just like Gabriel's Horn. The book referenced is on
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626:{\displaystyle A=2\pi \int _{1}^{a}{\frac {\sqrt {1+{\frac {1}{x^{4}}}}}{x}}\mathrm {d} x=2\pi \ln a+\pi \left_{1}^{a}}
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page might help you understand the principle of a solid of revolution and grasp the explanation more easily. -
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expert to forget that there are plenty of people who aren't able to follow the steps that are obvious to you.
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gravity. (plus, in a more humorous note, Gabriel was therefore blowing the horn before he even existed.) --
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have an infinite volume! The paint would dam itself at some narrow point and then fill from that point up.
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and Brian E. Blank's 2006 book on single variable calculus uses that name (on page 620), for example.
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may go on and conjuncture that 'either of the infinite area-volume pairs must have irrational value'.
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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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and the surface area is a bit like the third series while the volume is like the fourth. --
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Second, "you can show this by doing the calculations" is not useful for someone for whom
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Of course you will have trouble evaluating that expression in brackets at 1. --
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First, your summary is helpful, but no such summary exists on the main page.
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2\pi \int _{1}^{a}{\frac {\sqrt {1}}{x}}\ \mathrm {d} x=2\pi \ln a}" /: -->
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2\pi \int _{1}^{a}{\frac {\sqrt {1}}{x}}\ \mathrm {d} x=2\pi \ln a}": -->
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2\pi \int _{1}^{a}{\frac {\sqrt {1}}{x}}\ \mathrm {d} x=2\pi \ln a}
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So what I'm looking for is a plain-English explanation, I guess.
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is not well written and would greatly benefit from rewriting.
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Oeuvres complètes de Christiaan Huygens (1888), page 167-168
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emptied it, would it be painted?" In theory? Well, both yes
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is obviously infinite (you are adding 1 each time) and so a
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Why? It looks to me like or 00:35, 24 January 2008 (UTC)
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discussion also throws doubt on the claim of finite area.
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Hans Niels Jahnke: A history of analysis, pages 60-61
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2023:Christopher Lawrence Simpson
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1552:
1551:
1546:
1531:
1529:
1528:
1523:
1521:
1513:
1504:
1502:
1501:
1496:
1494:
1492:
1491:
1479:
1474:
1473:
1451:
1449:
1448:
1443:
1441:
1433:
1413:
1411:
1410:
1405:
1382:
1380:
1379:
1374:
1357:
1352:
1344:
1341:
1336:
1303:
1301:
1300:
1295:
1285:
1277:
1275:
1274:
1269:
1259:
1174:
1171:
1165:
1159:
1137:
1135:
1134:
1129:
1123:
1122:
1112:
1105:
1103:
1102:
1097:
1089:
1081:
1076:
1068:
1063:
1055:
1050:
1042:
1037:
1029:
1014:
1012:
1011:
1006:
998:
990:
985:
977:
972:
964:
959:
951:
946:
938:
924:
922:
921:
916:
908:
900:
895:
887:
882:
874:
869:
861:
856:
848:
837:divergent series
834:
832:
831:
826:
818:
810:
805:
797:
792:
784:
779:
771:
766:
758:
660:
632:
630:
629:
624:
621:
616:
611:
607:
606:
604:
602:
601:
589:
581:
576:
572:
571:
569:
567:
566:
554:
546:
494:
489:
483:
481:
480:
468:
460:
459:
456:
451:
417:
414:
413:
407:
384:
378:
377:
368:
367:
364:
359:
338:
333:
327:
325:
324:
312:
304:
303:
300:
295:
120:
119:
116:
113:
110:
89:
84:
83:
73:
66:
65:
55:
48:
31:
25:
24:
16:
2401:
2400:
2396:
2395:
2394:
2392:
2391:
2390:
2371:
2370:
2351:
2304:
2285:
2262:
2193:
2192:
2173:
2172:
2147:
2146:
2098:
2088:
2073:
2068:
2067:
2050:109.155.174.230
2034:
1937:
1932:
1911:
1858:
1854:
1824:
1820:
1798:
1794:
1793:
1789:
1753:
1749:
1722:
1646:
1645:
1606:
1605:
1576:
1557:
1556:
1534:
1533:
1507:
1506:
1483:
1465:
1454:
1453:
1421:
1420:
1396:
1395:
1317:
1316:
1281:
1280:
1255:
1254:
1235:
1226:
1212:
1204:
1169:
1163:
1157:
1154:
1114:
1108:
1107:
1023:
1022:
1017:harmonic series
932:
931:
842:
841:
752:
751:
683:
593:
558:
538:
534:
527:
523:
472:
426:
425:
316:
269:
268:
262:
189:
153:
117:
114:
111:
108:
107:
85:
78:
32:on Knowledge's
29:
12:
11:
5:
2399:
2397:
2389:
2388:
2383:
2373:
2372:
2350:
2347:
2303:
2300:
2289:EpsilonCarinae
2284:
2281:
2261:
2258:
2257:
2256:
2255:
2254:
2230:
2229:
2206:
2203:
2200:
2180:
2160:
2157:
2154:
2132:Olli Niemitalo
2107:
2104:
2101:
2095:
2091:
2085:
2080:
2076:
2046:Stack Exchange
2033:
2030:
2029:
2028:
2027:
2026:
2025:
2024:
2001:
2000:
1980:
1979:
1971:
1966:
1965:
1931:
1928:
1910:
1907:
1893:
1892:
1891:
1890:
1878:
1872:
1869:
1864:
1861:
1857:
1853:
1850:
1846:
1841:
1835:
1832:
1827:
1823:
1819:
1815:
1809:
1806:
1801:
1797:
1792:
1788:
1785:
1780:
1775:
1770:
1764:
1761:
1756:
1752:
1747:
1744:
1740:
1737:
1729:
1725:
1721:
1714:
1709:
1705:
1701:
1698:
1695:
1690:
1687:
1684:
1679:
1676:
1673:
1669:
1665:
1662:
1659:
1656:
1653:
1627:
1624:
1618:
1615:
1593:
1590:
1583:
1579:
1575:
1570:
1567:
1564:
1544:
1541:
1519:
1516:
1490:
1486:
1482:
1477:
1472:
1468:
1464:
1461:
1439:
1436:
1431:
1428:
1417:
1416:
1403:
1386:
1385:
1384:
1383:
1372:
1369:
1366:
1363:
1360:
1356:
1350:
1347:
1340:
1335:
1331:
1327:
1324:
1311:
1310:
1293:
1289:
1267:
1263:
1234:
1231:
1225:
1222:
1211:
1208:
1203:
1200:
1185:
1184:
1183:
1182:
1181:
1180:
1179:
1178:
1152:
1140:
1139:
1138:
1127:
1121:
1117:
1095:
1092:
1087:
1084:
1079:
1074:
1071:
1066:
1061:
1058:
1053:
1048:
1045:
1040:
1035:
1032:
1020:
1004:
1001:
996:
993:
988:
983:
980:
975:
970:
967:
962:
957:
954:
949:
944:
941:
929:
914:
911:
906:
903:
898:
893:
890:
885:
880:
877:
872:
867:
864:
859:
854:
851:
839:
824:
821:
816:
813:
808:
803:
800:
795:
790:
787:
782:
777:
774:
769:
764:
761:
741:
740:
739:
738:
730:
727:
723:
715:
712:
705:
704:
682:
679:
678:
677:
676:
675:
654:
653:
634:
633:
620:
615:
610:
600:
596:
592:
587:
584:
579:
575:
565:
561:
557:
552:
549:
544:
541:
537:
533:
530:
526:
521:
518:
515:
512:
509:
506:
503:
500:
497:
493:
487:
479:
475:
471:
466:
463:
455:
450:
446:
442:
439:
436:
433:
419:
418:
405:
402:
399:
396:
393:
390:
387:
383:
375:
371:
363:
358:
354:
350:
347:
344:
341:
337:
331:
323:
319:
315:
310:
307:
299:
294:
290:
286:
283:
280:
277:
261:
260:Exact integral
258:
257:
256:
233:
232:
221:
220:
219:
218:
207:
206:
188:
185:
163:
152:
149:
146:
145:
142:
141:
138:
137:
130:
124:
123:
121:
104:the discussion
91:
90:
74:
62:
61:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
2398:
2387:
2384:
2382:
2379:
2378:
2376:
2369:
2368:
2364:
2360:
2356:
2348:
2346:
2345:
2341:
2337:
2333:
2331:
2327:
2321:
2319:
2314:
2308:
2301:
2299:
2298:
2294:
2290:
2282:
2280:
2279:
2275:
2271:
2267:
2259:
2253:
2249:
2245:
2241:
2240:
2239:
2238:
2234:
2228:
2224:
2220:
2204:
2201:
2198:
2178:
2158:
2155:
2152:
2144:
2143:
2142:
2141:
2137:
2133:
2129:
2125:
2105:
2102:
2099:
2093:
2089:
2083:
2078:
2074:
2065:
2060:
2059:
2055:
2051:
2047:
2043:
2039:
2031:
2022:
2018:
2014:
2009:
2008:
2005:
2004:
2003:
2002:
1999:
1995:
1991:
1987:
1982:
1981:
1977:
1972:
1968:
1967:
1962:
1961:
1960:
1957:
1953:
1949:
1945:
1941:
1929:
1927:
1926:
1922:
1918:
1908:
1906:
1905:
1901:
1897:
1876:
1870:
1867:
1862:
1859:
1855:
1851:
1848:
1844:
1839:
1833:
1830:
1825:
1821:
1817:
1813:
1807:
1804:
1799:
1795:
1790:
1786:
1783:
1778:
1773:
1768:
1762:
1759:
1754:
1750:
1745:
1742:
1738:
1735:
1727:
1723:
1719:
1712:
1707:
1703:
1699:
1696:
1693:
1688:
1685:
1682:
1677:
1674:
1671:
1667:
1663:
1657:
1651:
1644:
1643:
1642:
1641:
1640:
1625:
1622:
1616:
1613:
1591:
1588:
1581:
1577:
1573:
1568:
1565:
1562:
1542:
1539:
1517:
1514:
1488:
1484:
1480:
1475:
1470:
1466:
1462:
1459:
1437:
1434:
1429:
1426:
1415:
1401:
1391:
1388:
1387:
1370:
1367:
1364:
1361:
1358:
1348:
1345:
1338:
1333:
1329:
1325:
1322:
1315:
1314:
1313:
1312:
1309:
1305:
1291:
1287:
1265:
1261:
1251:
1250:
1249:
1248:
1244:
1240:
1232:
1230:
1223:
1221:
1220:
1217:
1209:
1207:
1201:
1199:
1198:
1195:
1191:
1177:
1172:
1170:Contributions
1166:
1160:
1153:
1150:
1149:
1148:
1145:
1141:
1125:
1119:
1115:
1093:
1090:
1085:
1082:
1077:
1072:
1069:
1064:
1059:
1056:
1051:
1046:
1043:
1038:
1033:
1030:
1021:
1018:
1002:
999:
994:
991:
986:
981:
978:
973:
968:
965:
960:
955:
952:
947:
942:
939:
930:
928:
912:
909:
904:
901:
896:
891:
888:
883:
878:
875:
870:
865:
862:
857:
852:
849:
840:
838:
822:
819:
814:
811:
806:
801:
798:
793:
788:
785:
780:
775:
772:
767:
762:
759:
750:
749:
747:
746:
745:
744:
743:
742:
737:
734:
731:
728:
724:
720:
716:
713:
709:
708:
707:
706:
703:
700:
695:
694:
693:
692:
689:
680:
672:
668:
667:81.159.28.152
664:
658:
657:
656:
655:
652:
649:
645:
644:
643:
642:
639:
618:
613:
608:
598:
594:
590:
585:
582:
577:
573:
563:
559:
555:
550:
547:
542:
539:
535:
531:
528:
524:
519:
516:
513:
510:
507:
504:
501:
498:
495:
485:
477:
473:
469:
464:
461:
453:
448:
444:
440:
437:
434:
431:
424:
423:
422:
403:
400:
397:
394:
391:
388:
385:
373:
369:
361:
356:
352:
348:
345:
342:
339:
329:
321:
317:
313:
308:
305:
297:
292:
288:
284:
281:
278:
275:
267:
266:
265:
259:
254:
250:
245:
244:
243:
242:
239:
231:
228:
223:
222:
216:
211:
210:
209:
208:
204:
199:
198:
197:
195:
186:
184:
182:
178:
174:
170:
161:
159:
150:
135:
129:
126:
125:
122:
105:
101:
97:
96:
88:
82:
77:
75:
72:
68:
67:
63:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
2354:
2353:The section
2352:
2334:
2329:
2325:
2322:
2312:
2309:
2305:
2286:
2263:
2061:
2042:Google Books
2035:
1985:
1975:
1933:
1912:
1894:
1418:
1393:
1389:
1307:
1252:
1236:
1227:
1224:Minor change
1213:
1205:
1186:
684:
635:
420:
263:
238:74.134.8.244
234:
190:
167:— Preceding
162:
154:
93:
40:WikiProjects
1938:—Preceding
681:Readability
661:—Preceding
151:Not paradox
109:Mathematics
100:mathematics
59:Mathematics
2375:Categories
2270:Paradoctor
1944:Pandamonia
1304:from 1 to
2266:elsewhere
1194:hitman012
1158:Septegram
733:Septegram
726:infinite.
688:Septegram
227:hitman012
158:Llamatron
1990:Racerx11
1964:circles!
1952:contribs
1940:unsigned
1917:SGBailey
1909:Diagrams
663:unsigned
169:unsigned
2244:Rumping
2219:Rumping
1239:Rumping
1216:Bilgrau
1210:History
1144:Henrygb
699:Henrygb
638:Henrygb
215:Henrygb
30:C-class
1896:CiaPan
1604:, not
1505:, not
648:Spoon!
36:scale.
2326:right
2302:Title
1930:Paint
1015:(the
719:runes
343:: -->
249:~rAGU
2363:talk
2340:talk
2330:have
2293:talk
2274:talk
2248:talk
2223:talk
2136:talk
2054:talk
2017:talk
1994:talk
1948:talk
1921:talk
1900:talk
1243:talk
1190:this
1164:Talk
671:talk
253:talk
194:Matt
177:talk
1986:and
1976:did
203:toh
128:???
2377::
2365:)
2342:)
2295:)
2276:)
2268:.
2250:)
2225:)
2138:)
2103:−
2056:)
2019:)
1996:)
1954:)
1950:•
1923:)
1902:)
1863:−
1852:π
1826:−
1818:−
1800:−
1787:π
1755:−
1746:π
1720:π
1704:∫
1668:∫
1574:π
1481:π
1463:⋅
1460:π
1402:π
1368:
1365:ln
1330:∫
1245:)
1116:π
1094:⋯
1086:25
1073:16
1003:⋯
913:⋯
905:16
823:⋯
673:)
636:--
578:−
532:
529:ln
520:π
511:
508:ln
505:π
445:∫
441:π
401:
398:ln
395:π
353:∫
349:π
289:∫
285:π
179:)
2361:(
2338:(
2291:(
2272:(
2246:(
2221:(
2205:0
2202:=
2199:x
2179:y
2159:1
2156:=
2153:x
2134:(
2106:x
2100:1
2094:3
2090:x
2084:=
2079:2
2075:y
2052:(
2015:(
1992:(
1946:(
1919:(
1898:(
1877:)
1871:a
1868:1
1860:1
1856:(
1849:=
1845:]
1840:)
1834:1
1831:1
1822:(
1814:)
1808:a
1805:1
1796:(
1791:[
1784:=
1779:a
1774:1
1769:]
1763:x
1760:1
1751:[
1743:=
1739:x
1736:d
1728:2
1724:x
1713:a
1708:1
1700:=
1697:V
1694:d
1689:a
1686:=
1683:x
1678:1
1675:=
1672:x
1664:=
1661:)
1658:a
1655:(
1652:V
1626:x
1623:d
1617:x
1614:1
1592:x
1589:d
1582:2
1578:x
1569:=
1566:V
1563:d
1543:x
1540:d
1518:x
1515:1
1489:2
1485:x
1476:=
1471:2
1467:r
1438:x
1435:1
1430:=
1427:r
1414:.
1392:a
1371:a
1362:=
1359:x
1355:d
1349:x
1346:1
1339:a
1334:1
1326:=
1323:V
1308:.
1306:a
1292:x
1288:1
1266:x
1262:1
1241:(
1173:*
1167:*
1161:*
1155:*
1126:6
1120:2
1091:+
1083:1
1078:+
1070:1
1065:+
1060:9
1057:1
1052:+
1047:4
1044:1
1039:+
1034:1
1031:1
1000:+
995:5
992:1
987:+
982:4
979:1
974:+
969:3
966:1
961:+
956:2
953:1
948:+
943:1
940:1
910:+
902:1
897:+
892:8
889:1
884:+
879:4
876:1
871:+
866:2
863:1
858:+
853:1
850:1
820:+
815:1
812:1
807:+
802:1
799:1
794:+
789:1
786:1
781:+
776:1
773:1
768:+
763:1
760:1
669:(
619:a
614:1
609:]
599:4
595:x
591:1
586:+
583:1
574:)
564:4
560:x
556:1
551:+
548:1
543:+
540:1
536:(
525:[
517:+
514:a
502:2
499:=
496:x
492:d
486:x
478:4
474:x
470:1
465:+
462:1
454:a
449:1
438:2
435:=
432:A
404:a
392:2
389:=
386:x
382:d
374:x
370:1
362:a
357:1
346:2
340:x
336:d
330:x
322:4
318:x
314:1
309:+
306:1
298:a
293:1
282:2
279:=
276:A
255:)
251:(
175:(
136:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
