751:). If it is applied on a quadratic function, then it yields the trapezoid figure from Carlyle's solution to Leslie's problem (see graphic) with one of its sides being the diameter of the Carlyle circle. In an article from 1925 G. A. Miller pointed out that a slight modification of Lill's method applied to a normed quadratic function yields a circle that allows the geometric construction of the roots of that function and gave the explicit modern definition of what was later to be called Carlyle circle.
698:
230:
218:
68:
1184:
725:(1795–1881). However while the description in Leslie's book contains an analogous circle construction, it was presented solely in elementary geometric terms without the notion of a Cartesian coordinate system or a quadratic function and its roots:
1395:
1303:
681:. However there are practical problems for the implementation of the procedure; for example, it requires the construction of the Carlyle circle for the solution of the quadratic equation
433:
The
Carlyle circle associated with this quadratic has a diameter with endpoints at (0, 1) and (−1, −1) and center at (−1/2, 0). Carlyle circles are used to construct
1263:
701:
Carlyle's solution to Leslie's problem. The black line segment is divided in two segments in such a way that the two segments form a rectangle (green) being of equal
242:
1318:
1087:
971:[Graphical solution of numerical equations of all degrees having a single unknown, and description of an instrument invented for this purpose].
1279:
870:
1067:
815:
729:
To divide a straight line, whether internally or externally, so that the rectangle under its segments shall be equivalent to a given rectangle.
1015:
779:
1332:
969:"Résolution graphique des équations numériques de tous degrés à une seule inconnue, et description d'un instrument inventé dans ce but"
661:
using
Carlyle circles, as many as 24 Carlyle circles are to be constructed. One of these is the circle to solve the quadratic equation
798:. Ladislav Beran described in 1999 how the Carlyle circle can be used to construct the complex roots of a normed quadratic function.
1480:
1311:
1255:
1238:
1475:
1097:
589: − 1 = 0. Mark its intersection with the horizontal line (inside the original circle) as the point
52:
1358:
1433:
1092:
1060:
142:
The defining property of the
Carlyle circle can be established thus: the equation of the circle having the line segment
1485:
1465:
1143:
287:(1, 0). Removing the factor corresponding to this root, the other roots turn out to be roots of the equation
1295:
1133:
968:
229:
1424:
1228:
540:
Draw a horizontal line through the center of the circle. Mark one intersection with the circle as point
1495:
1373:
1325:
1271:
1053:
717:(1766–1832) described the geometric construction of roots of a quadratic equation with a circle in his book
714:
758:(1953) and pointed out the connection to Leslie and Carlyle. Later publications started to adopt the names
217:
1173:
241:
901:
1287:
1138:
957:). Note that the comment about Carlyle is not contained in earlier editions of the book (1809, 1811).
47:
of the quadratic equation are the horizontal coordinates of the intersections of the circle with the
1414:
1348:
1123:
697:
951:
Elements of geometry and plane trigonometry: With an appendix, and copious notes and illustrations
1490:
1408:
1118:
893:
186:
36:
1470:
1158:
1128:
1113:
1011:
1019:
547:
Construct a vertical line through the center. Mark one intersection with the circle as point
1368:
1163:
885:
748:
397:
107:
44:
32:
1247:
56:
48:
396: = −1. (These can be quickly shown to be true by direct substitution into the
1363:
1223:
1213:
1208:
1076:
722:
40:
1459:
1153:
636:
The fifth vertex is the intersection of the horizontal axis with the original circle.
1378:
1218:
1198:
787:
646:
234:
111:
835:
521:
This detailed procedure involving
Carlyle circles for the construction of regular
954:
677:
There is a procedure involving
Carlyle circles for the construction of a regular
1168:
744:
710:
20:
1419:
1148:
820:. The College Mathematics Journal, Vol. 21, No. 5 (Nov., 1990), pp. 362–369 (
778:) is used as well. DeTemple used in 1989 and 1991 Carlyle circles to devise
754:
Eves used the circle in the modern sense in one of the exercises of his book
1444:
1402:
1203:
1183:
795:
678:
633:. It intersects the original circle at two of the vertices of the pentagon.
622:. It intersects the original circle at two of the vertices of the pentagon.
67:
783:
645:
There is a similar method involving
Carlyle circles to construct regular
559:
522:
259:
222:
178:
1035:. The Mathematical Gazette, Vol. 83, No. 497 (Jul., 1999), pp. 287–291 (
993:. The Mathematical Gazette, Vol. 12, No. 179 (Dec., 1925), pp. 500–501 (
1036:
994:
918:
897:
821:
791:
658:
262:
is equivalent to the problem of constructing the roots of the equation
246:
310:
These roots can be represented in the form ω, ω, ω, ω where ω = exp (2
747:
published a graphical method to determine the roots of a polynomial (
530:
28:
889:
1439:
696:
1045:
871:"Carlyle circles and Lemoine simplicity of polygon constructions"
533:
in which to inscribe the pentagon and mark the center point
1396:
Arrangement in Grey and Black, No. 2: Portrait of Thomas
Carlyle
953:. Archibald Constable & Co, 3. Ausgabe, 1817, pp. 176, 340 (
702:
1304:
History of
Friedrich II. of Prussia, Called Frederick the Great
1049:
721:
and noted that this idea was provided by his former student
817:
Geometrical and
Graphical Solutions of Quadratic Equations
937:. Holt, Rinehart and Winston, 3rd edition, 1969, p. 73
649:. The figure to the right illustrates the procedure.
593:
and its intersection outside the circle as the point
1264:
On Heroes, Hero-Worship, & the Heroic in
History
1387:
1341:
1237:
1191:
1106:
933:See for instance Hornsby, DeTemple or Howard Eves:
1006:Rainer Kaenders (ed.), Reinhard Schmidt (ed.):
935:An Introduction into the History of Mathematics
770:, though in German speaking countries the term
727:
1033:The Complex Roots of a Quadratic from a Circle
280: = 1 which corresponds to the point
193: = 0 in the equation of the circle)
181:of the points where the circle intersects the
1061:
8:
991:Geometric Solution of the Quadratic Equation
490:These are then used to construct the points
51:. Carlyle circles have been used to develop
16:Circle associated with a quadratic equation
1319:Occasional Discourse on the Negro Question
1068:
1054:
1046:
864:
862:
860:
858:
856:
756:Introduction to the History of Mathematics
418:are the roots of the quadratic equation
71:Carlyle circle of the quadratic equation
1010:. Springer Spektrum, 2nd edition, 2014,
945:
943:
782:for regular polygons, in particular the
400:above and noting that ω = ω, and ω = ω.)
317:/5). Let these correspond to the points
240:
228:
216:
66:
807:
43:. The circle has the property that the
1280:Oliver Cromwell's Letters and Speeches
1008:Mit GeoGebra mehr Mathematik verstehen
929:
927:
780:compass-and-straightedge constructions
258:The problem of constructing a regular
840:From MathWorld—A Wolfram Web Resource
669: − 64 = 0.
189:of the equation (obtained by setting
7:
689: − 2 = 0.
429: − 1 = 0.
973:Nouvelles Annales de Mathématiques
14:
1312:Critical and Miscellaneous Essays
878:The American Mathematical Monthly
705:to another given rectangle (red).
581:. This is the Carlyle circle for
1256:The French Revolution: A History
1182:
213:Construction of regular polygons
869:DeTemple, Duane W. (Feb 1991).
713:(1911–2004), the mathematician
53:ruler-and-compass constructions
743:In 1867 the Austrian engineer
355: = ω + ω,
130:) as a diameter is called the
1:
1359:Condition-of-England Question
306: + 1 = 0.
273:One root of this equation is
269: − 1 = 0.
87:Given the quadratic equation
1434:Thomas Carlyle and His Works
362: = ω + ω
1333:Correspondence with Emerson
134:of the quadratic equation.
1514:
1144:Johann Wolfgang von Goethe
975:. 2nd series (in French).
573:Draw a circle centered at
482: = 2 cos(4
471: = 2 cos(2
447:. From the definitions of
245:Construction of a regular
233:Construction of a regular
1399:by James McNeill Whistler
1296:The Life of John Sterling
1180:
1083:
1481:Euclidean plane geometry
1436:" by Henry David Thoreau
625:Draw a circle of radius
614:Draw a circle of radius
221:Construction of regular
1374:Natural Supernaturalism
657:To construct a regular
597:. These are the points
1476:Constructible polygons
814:E. John Hornsby, Jr.:
741:
706:
250:
238:
226:
84:
700:
461:it also follows that
249:using Carlyle circles
244:
237:using Carlyle circles
232:
225:using Carlyle circles
220:
70:
1288:Latter-Day Pamphlets
1139:James Anthony Froude
955:online copy (Google)
738:, prop. XVII, p. 176
736:Elements of Geometry
719:Elements of Geometry
641:Regular heptadecagon
554:Construct the point
39:; it is named after
1349:Captain of Industry
1124:Ralph Waldo Emerson
834:Weisstein, Eric W.
200: −
114:joining the points
94: −
1486:Elementary algebra
1466:Euclidean geometry
1409:Laborare est Orare
1119:Jane Welsh Carlyle
707:
577:through the point
251:
239:
227:
106:the circle in the
85:
37:quadratic equation
35:associated with a
1453:
1452:
1159:Kitty Kirkpatrick
1129:Francis Espinasse
1114:William Allingham
1016:978-3-658-04222-6
967:Lill, E. (1867).
768:Carlyle algorithm
673:Regular 65537-gon
138:Defining property
118:(0, 1) and
1503:
1369:Great Man theory
1272:Past and Present
1186:
1164:John Stuart Mill
1070:
1063:
1056:
1047:
1040:
1031:Ladislav Beran:
1029:
1023:
1004:
998:
987:
981:
980:
964:
958:
947:
938:
931:
922:
916:
914:
912:
906:
900:. Archived from
875:
866:
851:
850:
848:
846:
836:"Carlyle Circle"
831:
825:
812:
739:
611:mentioned above.
525:is given below.
485:
474:
316:
254:Regular pentagon
173:) = 0.
165: − 1)(
108:coordinate plane
57:regular polygons
33:coordinate plane
1513:
1512:
1506:
1505:
1504:
1502:
1501:
1500:
1456:
1455:
1454:
1449:
1383:
1337:
1248:Sartor Resartus
1233:
1187:
1178:
1102:
1079:
1074:
1044:
1043:
1030:
1026:
1005:
1001:
988:
984:
966:
965:
961:
948:
941:
932:
925:
910:
908:
904:
890:10.2307/2323939
873:
868:
867:
854:
844:
842:
833:
832:
828:
813:
809:
804:
740:
733:
695:
675:
655:
653:Regular 257-gon
643:
610:
603:
517:
510:
503:
496:
483:
481:
472:
470:
460:
453:
446:
439:
417:
410:
395:
389:
382:
375:
361:
354:
344:
337:
330:
323:
314:
286:
279:
256:
215:
208: = 0.
161:) + (
146:as diameter is
140:
83: = 0.
65:
49:horizontal axis
17:
12:
11:
5:
1511:
1510:
1507:
1499:
1498:
1496:Thomas Carlyle
1493:
1488:
1483:
1478:
1473:
1468:
1458:
1457:
1451:
1450:
1448:
1447:
1442:
1437:
1430:
1426:Thomas Carlyle
1422:
1417:
1412:
1405:
1400:
1391:
1389:
1385:
1384:
1382:
1381:
1376:
1371:
1366:
1364:Dismal Science
1361:
1356:
1354:Carlyle circle
1351:
1345:
1343:
1339:
1338:
1336:
1335:
1330:
1322:
1315:
1308:
1300:
1292:
1284:
1276:
1268:
1260:
1252:
1243:
1241:
1235:
1234:
1232:
1231:
1226:
1224:London Library
1221:
1216:
1214:Craigenputtock
1211:
1209:21 Comely Bank
1206:
1201:
1195:
1193:
1189:
1188:
1181:
1179:
1177:
1176:
1171:
1166:
1161:
1156:
1151:
1146:
1141:
1136:
1131:
1126:
1121:
1116:
1110:
1108:
1104:
1103:
1101:
1100:
1095:
1090:
1084:
1081:
1080:
1077:Thomas Carlyle
1075:
1073:
1072:
1065:
1058:
1050:
1042:
1041:
1024:
999:
989:G. A. Miller:
982:
959:
939:
923:
852:
826:
806:
805:
803:
800:
764:Carlyle method
760:Carlyle circle
731:
723:Thomas Carlyle
694:
691:
674:
671:
654:
651:
642:
639:
638:
637:
634:
623:
612:
608:
601:
571:
552:
545:
538:
515:
508:
501:
494:
488:
487:
479:
468:
458:
451:
444:
437:
431:
430:
415:
408:
402:
401:
393:
387:
380:
373:
364:
363:
359:
352:
342:
335:
328:
321:
308:
307:
284:
277:
271:
270:
255:
252:
214:
211:
210:
209:
185:-axis are the
175:
174:
139:
136:
132:Carlyle circle
104:
103:
102: = 0
64:
61:
41:Thomas Carlyle
25:Carlyle circle
15:
13:
10:
9:
6:
4:
3:
2:
1509:
1508:
1497:
1494:
1492:
1489:
1487:
1484:
1482:
1479:
1477:
1474:
1472:
1469:
1467:
1464:
1463:
1461:
1446:
1443:
1441:
1438:
1435:
1431:
1429:
1427:
1423:
1421:
1418:
1416:
1413:
1411:
1410:
1406:
1404:
1401:
1398:
1397:
1393:
1392:
1390:
1386:
1380:
1377:
1375:
1372:
1370:
1367:
1365:
1362:
1360:
1357:
1355:
1352:
1350:
1347:
1346:
1344:
1340:
1334:
1331:
1328:
1327:
1326:Reminiscences
1323:
1320:
1316:
1314:
1313:
1309:
1306:
1305:
1301:
1298:
1297:
1293:
1290:
1289:
1285:
1282:
1281:
1277:
1274:
1273:
1269:
1266:
1265:
1261:
1258:
1257:
1253:
1250:
1249:
1245:
1244:
1242:
1240:
1236:
1230:
1227:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1207:
1205:
1202:
1200:
1197:
1196:
1194:
1190:
1185:
1175:
1174:John Sterling
1172:
1170:
1167:
1165:
1162:
1160:
1157:
1155:
1154:Edward Irving
1152:
1150:
1147:
1145:
1142:
1140:
1137:
1135:
1132:
1130:
1127:
1125:
1122:
1120:
1117:
1115:
1112:
1111:
1109:
1105:
1099:
1096:
1094:
1091:
1089:
1086:
1085:
1082:
1078:
1071:
1066:
1064:
1059:
1057:
1052:
1051:
1048:
1038:
1034:
1028:
1025:
1021:
1017:
1013:
1009:
1003:
1000:
996:
992:
986:
983:
978:
974:
970:
963:
960:
956:
952:
949:John Leslie:
946:
944:
940:
936:
930:
928:
924:
920:
907:on 2015-12-21
903:
899:
895:
891:
887:
884:(2): 97–208.
883:
879:
872:
865:
863:
861:
859:
857:
853:
841:
837:
830:
827:
823:
819:
818:
811:
808:
801:
799:
797:
793:
789:
785:
781:
777:
773:
769:
765:
761:
757:
752:
750:
749:Lill's method
746:
737:
734:John Leslie,
730:
726:
724:
720:
716:
712:
709:According to
704:
699:
692:
690:
688:
685: +
684:
680:
672:
670:
668:
665: +
664:
660:
652:
650:
648:
647:heptadecagons
640:
635:
632:
628:
624:
621:
617:
613:
607:
600:
596:
592:
588:
585: +
584:
580:
576:
572:
569:
565:
561:
557:
553:
550:
546:
543:
539:
536:
532:
528:
527:
526:
524:
519:
514:
507:
500:
493:
478:
467:
464:
463:
462:
457:
450:
443:
436:
428:
425: +
424:
421:
420:
419:
414:
407:
399:
392:
386:
379:
376: +
372:
369:
368:
367:
358:
351:
348:
347:
346:
341:
334:
327:
320:
313:
305:
302: +
301:
298: +
297:
294: +
293:
290:
289:
288:
283:
276:
268:
265:
264:
263:
261:
253:
248:
243:
236:
231:
224:
219:
212:
207:
204: +
203:
199:
196:
195:
194:
192:
188:
184:
180:
172:
169: −
168:
164:
160:
157: −
156:
152:
149:
148:
147:
145:
137:
135:
133:
129:
125:
121:
117:
113:
109:
101:
98: +
97:
93:
90:
89:
88:
82:
79: +
78:
75: −
74:
69:
62:
60:
58:
54:
50:
46:
42:
38:
34:
30:
27:is a certain
26:
22:
1425:
1407:
1394:
1379:Sage writing
1353:
1324:
1310:
1302:
1294:
1286:
1278:
1270:
1262:
1254:
1246:
1219:5 Cheyne Row
1199:Arched House
1134:John Forster
1032:
1027:
1007:
1002:
990:
985:
976:
972:
962:
950:
934:
909:. Retrieved
902:the original
881:
877:
843:. Retrieved
839:
829:
816:
810:
788:heptadecagon
775:
771:
767:
763:
759:
755:
753:
742:
735:
728:
718:
708:
686:
682:
676:
666:
662:
656:
644:
630:
626:
619:
615:
605:
598:
594:
590:
586:
582:
578:
574:
567:
563:
555:
548:
541:
534:
520:
512:
505:
498:
491:
489:
476:
465:
455:
448:
441:
434:
432:
426:
422:
412:
405:
403:
390:
384:
383:= −1,
377:
370:
365:
356:
349:
339:
332:
325:
318:
311:
309:
303:
299:
295:
291:
281:
274:
272:
266:
257:
235:heptadecagon
205:
201:
197:
190:
182:
176:
170:
166:
162:
158:
154:
150:
143:
141:
131:
127:
123:
119:
115:
112:line segment
105:
99:
95:
91:
86:
80:
76:
72:
24:
18:
1307:(1858–1865)
1169:John Ruskin
1098:Prose style
772:Lill circle
745:Eduard Lill
715:John Leslie
711:Howard Eves
629:and center
618:and center
345:. Letting
110:having the
21:mathematics
1460:Categories
1420:Smelfungus
1149:Leigh Hunt
1093:Philosophy
979:: 359–362.
911:6 November
802:References
776:Lill-Kreis
63:Definition
1491:Equations
1445:Yggdrasil
1428:(Millais)
1403:Dryasdust
1204:Scotsbrig
1088:Allusions
1022:(German)
796:65537-gon
679:65537-gon
523:pentagons
366:we have
179:abscissas
45:solutions
1471:Polygons
1321:" (1849)
794:and the
784:pentagon
732:—
560:midpoint
260:pentagon
223:pentagon
1415:Phoenix
1388:Related
898:2323939
792:257-gon
693:History
659:257-gon
558:as the
529:Draw a
398:quartic
247:257-gon
126:,
1329:(1881)
1299:(1851)
1291:(1850)
1283:(1845)
1275:(1843)
1267:(1841)
1259:(1837)
1251:(1831)
1229:Statue
1192:Places
1107:People
1018:, pp.
1014:
896:
845:21 May
790:, the
786:, the
531:circle
29:circle
1440:Vates
1342:Ideas
1239:Works
1037:JSTOR
1020:68-71
995:JSTOR
919:JSTOR
905:(PDF)
894:JSTOR
874:(PDF)
822:JSTOR
475:/5),
187:roots
31:in a
1012:ISBN
913:2011
847:2013
703:area
604:and
566:and
486:/5).
454:and
440:and
411:and
177:The
23:, a
886:doi
766:or
562:of
404:So
55:of
19:In
1462::
1039:)
942:^
926:^
892:.
882:98
880:.
876:.
855:^
838:.
762:,
627:OA
616:OA
518:.
511:,
504:,
497:,
338:,
331:,
324:,
202:sx
144:AB
96:sx
77:sx
59:.
1432:"
1317:"
1069:e
1062:t
1055:v
997:)
977:6
921:)
917:(
915:.
888::
849:.
824:)
774:(
687:x
683:x
667:x
663:x
631:V
620:W
609:2
606:p
602:1
599:p
595:V
591:W
587:x
583:x
579:A
575:M
570:.
568:B
564:O
556:M
551:.
549:A
544:.
542:B
537:.
535:O
516:4
513:P
509:3
506:P
502:2
499:P
495:1
492:P
484:π
480:2
477:p
473:π
469:1
466:p
459:2
456:p
452:1
449:p
445:2
442:p
438:1
435:p
427:x
423:x
416:2
413:p
409:1
406:p
394:2
391:p
388:1
385:p
381:2
378:p
374:1
371:p
360:2
357:p
353:1
350:p
343:4
340:P
336:3
333:P
329:2
326:P
322:1
319:P
315:π
312:i
304:z
300:z
296:z
292:z
285:0
282:P
278:0
275:z
267:z
206:p
198:x
191:y
183:x
171:p
167:y
163:y
159:s
155:x
153:(
151:x
128:p
124:s
122:(
120:B
116:A
100:p
92:x
81:p
73:x
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
