122:
184:
20:
416:
837:
771:
608:
697:
375:
328:
913:
281:
509:
To accommodate the case of negative logarithms and the corresponding negative hyperbolic angles, different hyperbolic sectors are constructed according to whether
1060:
483:
502:
producing a unit of area (under the hyperbola or in a hyperbolic sector in standard position). Then the natural logarithm could be recognized as the
869:. To establish hyperbolic measure on a line, Klein noted that the area of a hyperbolic sector provided visual illustration of the concept.
475:
addressed the problem of computing the areas bounded by a hyperbola. His findings led to the natural logarithm function, once called the
1037:
1057:
458:
467:
1081:
780:
714:
993:
516:
1086:
613:
997:
495:
472:
980:
454:
491:
343:
296:
915:. The area of such hyperbolic sectors has been used to define hyperbolic distance in a geometry textbook.
862:
81:
875:
198:
121:
31:
407:= 1 onto the main diagonal, in his article "Note on the addition theorem for hyperbolic functions".
1091:
866:
852:
334:
217:
110:
85:
392:
183:
1053:
1033:
432:
233:
138:
253:
952:
503:
400:
385:
191:
172:
1043:
1040:
1016:
924:
126:
35:
487:
446:
229:
175:
at the origin, with the measure of the latter being defined as the area of the former.
89:
43:
1076:
1070:
944:
965:
1008:
865:
was published in 1928, it provided a foundation for the subject by reference to
858:
19:
486:, the natural logarithm was known in terms of the area of a hyperbolic sector.
479:
since it is obtained by integrating, or finding the area, under the hyperbola.
471:), the hyperbolic quadrature required the invention in 1647 of a new function:
462:
245:
71:
39:
171:
When in standard position, a hyperbolic sector corresponds to a positive
156:, add triangle {(0, 0), (1, 0), (1, 1)}, and subtract triangle {(0, 0), (
513:
is greater or less than one. A variable right triangle with area 1/2 is
415:
970:, Chapter VI: "On the connection of common and hyperbolic trigonometry"
391:
The analogy between circular and hyperbolic functions was described by
80:, or the corresponding region when this hyperbola is re-scaled and its
286:
with the hypotenuse being the segment from the origin to the point (
839:
This convention is in accord with a negative natural logarithm for
414:
182:
120:
16:
Region of the
Cartesian plane bounded by a hyperbola and two radii
290:) on the hyperbola. The length of the base of this triangle is
134:
461:. Whereas quadrature of the parabola had been accomplished by
403:
used such triangles, projecting from a point on the hyperbola
129:, shown squeezing rectangles and rotating a hyperbolic sector
224:
When in standard position, a hyperbolic sector determines a
18:
1058:
On the introduction of the notion of hyperbolic functions
711:. A positive hyperbolic angle is given by the area of
872:
Hyperbolic sectors can also be drawn to the hyperbola
190:(yellow) and hyperbolic sector (red) corresponding to
878:
783:
717:
616:
519:
346:
299:
256:
949:
Ideas and
Methods of Affine and Projective Geometry
46:from the origin to it. For example, the two points
907:
831:
765:
691:
602:
369:
322:
275:
832:{\displaystyle \int _{x}^{1}{\frac {dt}{t}}+V-T.}
766:{\displaystyle \int _{1}^{x}{\frac {dt}{t}}+T-V.}
699:The natural logarithm is known as the area under
168:)} (both triangles of which have the same area).
457:of the hyperbola. The other cases are given by
137:of a hyperbolic sector in standard position is
92:. A hyperbolic sector in standard position has
603:{\displaystyle V=\{(x,1/x),\ (x,0),\ (0,0)\}.}
88:leaving the center at the origin, as with the
1061:Bulletin of the American Mathematical Society
8:
1013:Vorlesungen über Nicht-Euklidische Geometrie
773:A negative hyperbolic angle is given by the
692:{\displaystyle T=\{(1,1),\ (1,0),\ (0,0)\}.}
683:
623:
594:
526:
484:Introduction to the Analysis of the Infinite
955:), page 151, Ministry of Education, Moscow
897:
885:
877:
799:
793:
788:
782:
733:
727:
722:
716:
615:
541:
518:
366:
347:
345:
319:
300:
298:
272:
255:
109:Hyperbolic sectors are the basis for the
236:at the origin, base on the diagonal ray
936:
125:Hyperbolic sector area is preserved by
370:{\displaystyle {\sqrt {2}}\sinh u,\,}
323:{\displaystyle {\sqrt {2}}\cosh u,\,}
7:
218:hyperbolic cosine and sine functions
1030:Perspectives on Projective Geometry
908:{\displaystyle y={\sqrt {1+x^{2}}}}
482:Before 1748 and the publication of
506:to the transcendental function e.
14:
490:changed that when he introduced
209:). The legs of the triangle are
967:Trigonometry and Double Algebra
397:Trigonometry and Double Algebra
983:20:145–8, see diagram page 146
680:
668:
659:
647:
638:
626:
591:
579:
570:
558:
549:
529:
468:The Quadrature of the Parabola
459:Cavalieri's quadrature formula
1:
1028:Jürgen Richter-Gebert (2011)
494:such as 10. Euler identified
465:in the third century BC (in
1108:
964:Augustus De Morgan (1849)
850:
453:= –1 corresponding to the
430:
244:, and third vertex on the
998:Humboldt State University
994:The History of Logarithms
473:Gregoire de Saint-Vincent
148:Proof: Integrate under 1/
981:Messenger of Mathematics
979:William Burnside (1890)
492:transcendental functions
276:{\displaystyle xy=1,\,}
1015:, p. 173, figure 113,
909:
863:non-Euclidean geometry
833:
767:
693:
610:The isosceles case is
604:
428:
427:as exploited by Euler.
371:
324:
277:
221:
130:
23:
943:V.G. Ashkinuse &
910:
834:
768:
694:
605:
418:
372:
325:
278:
199:rectangular hyperbola
186:
124:
72:rectangular hyperbola
22:
876:
781:
715:
614:
517:
477:hyperbolic logarithm
411:Hyperbolic logarithm
344:
297:
254:
111:hyperbolic functions
1082:Elementary geometry
867:projective geometry
853:Hyperbolic geometry
847:Hyperbolic geometry
798:
732:
449:except in the case
437:It is known that f(
384:is the appropriate
226:hyperbolic triangle
188:Hyperbolic triangle
179:Hyperbolic triangle
905:
829:
784:
763:
718:
689:
600:
429:
393:Augustus De Morgan
367:
320:
273:
222:
131:
24:
1087:Integral calculus
1054:Mellen W. Haskell
903:
812:
746:
667:
646:
578:
557:
445:has an algebraic
433:Natural logarithm
352:
305:
139:natural logarithm
28:hyperbolic sector
1099:
1046:
1026:
1020:
1006:
1000:
992:Martin Flashman
990:
984:
977:
971:
962:
956:
941:
914:
912:
911:
906:
904:
902:
901:
886:
838:
836:
835:
830:
813:
808:
800:
797:
792:
772:
770:
769:
764:
747:
742:
734:
731:
726:
707:between one and
698:
696:
695:
690:
665:
644:
609:
607:
606:
601:
576:
555:
545:
504:inverse function
498:as the value of
401:William Burnside
386:hyperbolic angle
376:
374:
373:
368:
353:
348:
329:
327:
326:
321:
306:
301:
282:
280:
279:
274:
215:
214:
192:hyperbolic angle
173:hyperbolic angle
105:
98:
84:is altered by a
79:
69:
57:
1107:
1106:
1102:
1101:
1100:
1098:
1097:
1096:
1067:
1066:
1050:
1049:
1027:
1023:
1017:Julius Springer
1007:
1003:
991:
987:
978:
974:
963:
959:
942:
938:
933:
925:Squeeze mapping
921:
893:
874:
873:
855:
849:
801:
779:
778:
735:
713:
712:
612:
611:
515:
514:
435:
419:Unit area when
413:
342:
341:
295:
294:
252:
251:
212:
210:
181:
127:squeeze mapping
119:
100:
93:
74:
59:
47:
36:Cartesian plane
17:
12:
11:
5:
1105:
1103:
1095:
1094:
1089:
1084:
1079:
1069:
1068:
1065:
1064:
1048:
1047:
1021:
1001:
985:
972:
957:
935:
934:
932:
929:
928:
927:
920:
917:
900:
896:
892:
889:
884:
881:
851:Main article:
848:
845:
828:
825:
822:
819:
816:
811:
807:
804:
796:
791:
787:
762:
759:
756:
753:
750:
745:
741:
738:
730:
725:
721:
688:
685:
682:
679:
676:
673:
670:
664:
661:
658:
655:
652:
649:
643:
640:
637:
634:
631:
628:
625:
622:
619:
599:
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593:
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587:
584:
581:
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572:
569:
566:
563:
560:
554:
551:
548:
544:
540:
537:
534:
531:
528:
525:
522:
488:Leonhard Euler
447:antiderivative
431:Main article:
412:
409:
378:
377:
365:
362:
359:
356:
351:
331:
330:
318:
315:
312:
309:
304:
284:
283:
271:
268:
265:
262:
259:
230:right triangle
180:
177:
118:
115:
90:unit hyperbola
15:
13:
10:
9:
6:
4:
3:
2:
1104:
1093:
1090:
1088:
1085:
1083:
1080:
1078:
1075:
1074:
1072:
1062:
1059:
1055:
1052:
1051:
1045:
1042:
1039:
1038:9783642172854
1035:
1031:
1025:
1022:
1018:
1014:
1010:
1005:
1002:
999:
995:
989:
986:
982:
976:
973:
969:
968:
961:
958:
954:
950:
946:
940:
937:
930:
926:
923:
922:
918:
916:
898:
894:
890:
887:
882:
879:
870:
868:
864:
860:
854:
846:
844:
842:
826:
823:
820:
817:
814:
809:
805:
802:
794:
789:
785:
776:
760:
757:
754:
751:
748:
743:
739:
736:
728:
723:
719:
710:
706:
702:
686:
677:
674:
671:
662:
656:
653:
650:
641:
635:
632:
629:
620:
617:
597:
588:
585:
582:
573:
567:
564:
561:
552:
546:
542:
538:
535:
532:
523:
520:
512:
507:
505:
501:
497:
493:
489:
485:
480:
478:
474:
470:
469:
464:
460:
456:
452:
448:
444:
440:
434:
426:
422:
417:
410:
408:
406:
402:
398:
394:
389:
387:
383:
363:
360:
357:
354:
349:
340:
339:
338:
336:
316:
313:
310:
307:
302:
293:
292:
291:
289:
269:
266:
263:
260:
257:
250:
249:
248:
247:
243:
240: =
239:
235:
231:
227:
219:
208:
204:
200:
196:
193:
189:
185:
178:
176:
174:
169:
167:
163:
159:
155:
151:
146:
144:
140:
136:
128:
123:
116:
114:
112:
107:
103:
96:
91:
87:
83:
77:
73:
67:
63:
55:
51:
45:
41:
38:bounded by a
37:
33:
29:
21:
1029:
1024:
1012:
1004:
988:
975:
966:
960:
948:
945:Isaak Yaglom
939:
871:
856:
840:
777:of the area
774:
708:
704:
700:
510:
508:
499:
481:
476:
466:
450:
442:
438:
436:
424:
420:
404:
396:
390:
381:
379:
332:
287:
285:
241:
237:
225:
223:
206:
202:
194:
187:
170:
165:
161:
157:
153:
149:
147:
142:
132:
108:
101:
94:
75:
65:
61:
53:
49:
27:
25:
1063:1(6):155–9.
1009:Felix Klein
861:'s book on
859:Felix Klein
82:orientation
1092:Logarithms
1071:Categories
1032:, p. 385,
931:References
843:in (0,1).
463:Archimedes
455:quadrature
216:times the
201:(equation
197:, to the
152:from 1 to
821:−
786:∫
755:−
720:∫
358:
311:
246:hyperbola
232:with one
40:hyperbola
1019:, Berlin
919:See also
775:negative
399:(1849).
335:altitude
333:and the
86:rotation
70:on the
42:and two
1056:(1895)
1044:2791970
1011:(1928)
953:Russian
947:(1962)
395:in his
211:√
160:, 0), (
34:of the
1036:
666:
645:
577:
556:
380:where
234:vertex
228:, the
104:> 1
32:region
996:from
857:When
30:is a
1077:Area
1034:ISBN
951:(in
703:= 1/
441:) =
355:sinh
337:is
308:cosh
288:x, y
205:= 1/
164:, 1/
141:of
135:area
133:The
117:Area
99:and
64:, 1/
58:and
52:, 1/
44:rays
97:= 1
78:= 1
1073::
1041:MR
423:=
405:xy
388:.
145:.
113:.
106:.
76:xy
26:A
899:2
895:x
891:+
888:1
883:=
880:y
841:x
827:.
824:T
818:V
815:+
810:t
806:t
803:d
795:1
790:x
761:.
758:V
752:T
749:+
744:t
740:t
737:d
729:x
724:1
709:x
705:x
701:y
687:.
684:}
681:)
678:0
675:,
672:0
669:(
663:,
660:)
657:0
654:,
651:1
648:(
642:,
639:)
636:1
633:,
630:1
627:(
624:{
621:=
618:T
598:.
595:}
592:)
589:0
586:,
583:0
580:(
574:,
571:)
568:0
565:,
562:x
559:(
553:,
550:)
547:x
543:/
539:1
536:,
533:x
530:(
527:{
524:=
521:V
511:x
500:b
496:e
451:p
443:x
439:x
425:e
421:b
382:u
364:,
361:u
350:2
317:,
314:u
303:2
270:,
267:1
264:=
261:y
258:x
242:x
238:y
220:.
213:2
207:x
203:y
195:u
166:b
162:b
158:b
154:b
150:x
143:b
102:b
95:a
68:)
66:b
62:b
60:(
56:)
54:a
50:a
48:(
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