282:
102:
92:
Motivated by the work of
Schweikart, Taurinus examined the model of geometry on a "sphere" of imaginary radius, which he called "logarithmic-spherical" (now called hyperbolic geometry). He published his "theory of parallel lines" in 1825 and "Geometriae prima elementa" in 1826. For instance, in his
440:
Taurinus corresponded with Gauss about his ideas in 1824. In his reply, Gauss mentioned some of his own ideas on the subject, and encouraged
Taurinus to further investigate this topic, but he also told Taurinus not to publicly cite Gauss. When Taurinus sent his works to Gauss, the latter didn't
420:
Taurinus described his logarithmic-spherical geometry as the "third system" besides
Euclidean geometry and spherical geometry, and pointed out that infinitely many systems exist depending on an arbitrary constant. While he noticed that no contradictions can be found in his logarithmic-spherical
441:
respond – according to Stäckel that was probably due to the fact that
Taurinus mentioned Gauss in the prefaces of his books. In addition, Taurinus sent some copies of his "Geometriae prima elementa" to friends and authorities (Stäckel reported a positive reply by
429:, as well as Zacharias, Taurinus must be given credit as a founder of non-Euclidean trigonometry (together with Gauss), but his contributions cannot be considered as being on the same level as those of the main founders of non-Euclidean geometry,
277:{\displaystyle A=\operatorname {arccos} {\frac {\cos \left(\alpha {\sqrt {-1}}\right)-\cos \left(\beta {\sqrt {-1}}\right)\cos \left(\gamma {\sqrt {-1}}\right)}{\sin \left(\beta {\sqrt {-1}}\right)\sin \left(\gamma {\sqrt {-1}}\right)}}}
415:
331:
445:). Dissatisfied with the lack of recognition, Taurinus burnt the remaining copies of that book – the only copy found by Stäckel and Engel was in the library of the
343:
703:
81:) in which the parallel postulate is not satisfied, and in which the sum of three angles of a triangle is less than two right angles (which is now called
672:
85:). While Schweikart never published his work (which he called "astral geometry"), he sent a short summary of its main principles by letter to
561:
It contains excerpts from
Taurinus' "Theorie der Parallellinien" and a partial German translation of "Geometriae prima elementa".
426:
290:
74:
708:
449:. In 2015, another copy of the "Geometriae prima elementa" was digitized and made freely available online by the
94:
66:
677:
450:
78:
86:
30:
641:
42:
698:
693:
663:
430:
334:
82:
667:
446:
531:
Zeitschrift fĂĽr
Mathematik und Physik, Supplement, Abhandlungen zur Geschichte der Mathematik
601:
623:"Elementargeometrie und elementare nicht-Euklidische Geometrie in synthetischer Behandlung"
622:
434:
422:
495:
421:
geometry, he remained convinced of the special role of
Euclidean geometry. According to
41:
Franz
Taurinus was the son of Julius Ephraim Taurinus, a court official of the Count of
554:
410:{\displaystyle \cosh \alpha =\cosh \beta \cosh \gamma -\sinh \beta \sinh \gamma \cos A}
687:
606:
589:
572:
70:
26:
54:
475:
526:
46:
442:
73:, among other things about mathematics. Schweikart examined a model (after
50:
574:
Non-Euclidean geometry: A critical and historical study of its development
23:
497:
Geometriae prima elementa. Recensuit et novas observationes adjecit
93:"Geometriae prima elementa" on p. 66, Taurinus defined the
556:
Die
Theorie der Parallellinien von Euklid bis auf Gauss
346:
326:{\displaystyle \cos \left(\alpha {\sqrt {-1}}\right)}
293:
105:
409:
325:
276:
45:, and Luise Juliane Schweikart. He studied law in
627:Encyclopädie der mathematischen Wissenschaften
8:
590:"Non-euclidean geometry—A re-interpretation"
57:. He lived as a private scholar in Cologne.
22:(15 November 1794 – 13 February 1874) was a
605:
345:
308:
292:
256:
227:
196:
167:
135:
118:
104:
69:(1780–1859), who was a law professor in
673:MacTutor History of Mathematics Archive
512:
466:
489:
487:
559:. Leipzig: Teubner. pp. 267–286.
548:
546:
544:
520:
518:
516:
65:Taurinus corresponded with his uncle
7:
704:19th-century German mathematicians
14:
494:Taurinus, Franz Adolph (1826).
474:Taurinus, Franz Adolph (1825).
553:Engel, F; Stäckel, P. (1895).
1:
29:who is known for his work on
607:10.1016/0315-0860(79)90124-1
725:
477:Theorie der Parallellinien
75:Giovanni Girolamo Saccheri
95:hyperbolic law of cosines
67:Ferdinand Karl Schweikart
678:University of St Andrews
451:University of Regensburg
527:"Franz Adolph Taurinus"
79:Johann Heinrich Lambert
621:Zacharias, M. (1913).
577:. Chicago: Open Court.
411:
327:
278:
31:non-Euclidean geometry
412:
328:
279:
20:Franz Adolph Taurinus
664:Robertson, Edmund F.
640:Stäckel, P. (1917).
594:Historia Mathematica
525:Stäckel, P. (1899).
344:
335:hyperbolic functions
291:
103:
16:German mathematician
662:O'Connor, John J.;
642:"GauĂź als Geometer"
571:Bonola, R. (1912).
431:Nikolai Lobachevsky
87:Carl Friedrich GauĂź
83:hyperbolic geometry
61:Hyperbolic geometry
629:. 3.1.2: 862–1176.
447:University of Bonn
407:
337:, it has the form
323:
274:
588:Gray, J. (1979).
508:Secondary sources
462:Works of Taurinus
316:
272:
264:
235:
204:
175:
143:
716:
709:German geometers
680:
668:"Franz Taurinus"
650:
649:
637:
631:
630:
618:
612:
611:
609:
585:
579:
578:
568:
562:
560:
550:
539:
538:
522:
502:
501:
491:
482:
481:
471:
416:
414:
413:
408:
332:
330:
329:
324:
322:
318:
317:
309:
287:When solved for
283:
281:
280:
275:
273:
271:
270:
266:
265:
257:
241:
237:
236:
228:
211:
210:
206:
205:
197:
181:
177:
176:
168:
149:
145:
144:
136:
119:
43:Erbach-Schönberg
724:
723:
719:
718:
717:
715:
714:
713:
684:
683:
661:
658:
653:
639:
638:
634:
620:
619:
615:
587:
586:
582:
570:
569:
565:
552:
551:
542:
524:
523:
514:
510:
505:
500:. Köln: Bachem.
493:
492:
485:
480:. Köln: Bachem.
473:
472:
468:
464:
459:
427:Friedrich Engel
342:
341:
304:
300:
289:
288:
252:
248:
223:
219:
212:
192:
188:
163:
159:
131:
127:
120:
101:
100:
63:
39:
17:
12:
11:
5:
722:
720:
712:
711:
706:
701:
696:
686:
685:
682:
681:
657:
656:External links
654:
652:
651:
632:
613:
600:(3): 236–258.
580:
563:
540:
511:
509:
506:
504:
503:
483:
465:
463:
460:
458:
455:
418:
417:
406:
403:
400:
397:
394:
391:
388:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
349:
321:
315:
312:
307:
303:
299:
296:
285:
284:
269:
263:
260:
255:
251:
247:
244:
240:
234:
231:
226:
222:
218:
215:
209:
203:
200:
195:
191:
187:
184:
180:
174:
171:
166:
162:
158:
155:
152:
148:
142:
139:
134:
130:
126:
123:
117:
114:
111:
108:
62:
59:
38:
35:
15:
13:
10:
9:
6:
4:
3:
2:
721:
710:
707:
705:
702:
700:
697:
695:
692:
691:
689:
679:
675:
674:
669:
665:
660:
659:
655:
647:
643:
636:
633:
628:
624:
617:
614:
608:
603:
599:
595:
591:
584:
581:
576:
575:
567:
564:
558:
557:
549:
547:
545:
541:
536:
532:
528:
521:
519:
517:
513:
507:
499:
498:
490:
488:
484:
479:
478:
470:
467:
461:
456:
454:
452:
448:
444:
438:
436:
432:
428:
424:
404:
401:
398:
395:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
350:
347:
340:
339:
338:
336:
319:
313:
310:
305:
301:
297:
294:
267:
261:
258:
253:
249:
245:
242:
238:
232:
229:
224:
220:
216:
213:
207:
201:
198:
193:
189:
185:
182:
178:
172:
169:
164:
160:
156:
153:
150:
146:
140:
137:
132:
128:
124:
121:
115:
112:
109:
106:
99:
98:
97:
96:
90:
88:
84:
80:
76:
72:
68:
60:
58:
56:
52:
48:
44:
36:
34:
32:
28:
27:mathematician
25:
21:
671:
646:Gött. Nachr.
645:
635:
626:
616:
597:
593:
583:
573:
566:
555:
534:
530:
496:
476:
469:
439:
435:János Bolyai
423:Paul Stäckel
419:
286:
91:
64:
40:
19:
18:
699:1874 deaths
694:1794 births
688:Categories
537:: 401–427.
457:References
333:and using
71:Königsberg
47:Heidelberg
648:: 25–142.
443:Georg Ohm
402:
396:γ
393:
387:β
384:
378:−
375:γ
372:
366:β
363:
354:α
351:
311:−
306:α
298:
259:−
254:γ
246:
230:−
225:β
217:
199:−
194:γ
186:
170:−
165:β
157:
151:−
138:−
133:α
125:
116:
55:Göttingen
113:arccos
51:GieĂźen
24:German
433:and
425:and
390:sinh
381:sinh
369:cosh
360:cosh
348:cosh
77:and
53:and
37:Life
602:doi
399:cos
295:cos
243:sin
214:sin
183:cos
154:cos
122:cos
33:.
690::
676:,
670:,
666:,
644:.
625:.
596:.
592:.
543:^
535:44
533:.
529:.
515:^
486:^
453:.
437:.
89:.
49:,
610:.
604::
598:6
405:A
357:=
320:)
314:1
302:(
268:)
262:1
250:(
239:)
233:1
221:(
208:)
202:1
190:(
179:)
173:1
161:(
147:)
141:1
129:(
110:=
107:A
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.