146:(long side) of the triangle runs from this instrument to the landmark on the other side of the river. Then the surveyor constructs a second right-angled triangle, which is the same size as the first one. The hypotenuse of the second triangle continues the hypotenuse of the first triangle and is recorded with a mark on the land. One cathetus is the continuation of the path along the riverbank, the second cathetus goes inland, joining up with the hypotenuse of the second triangle. The length of this second cathetus will be equal to the width of the river.
181:
286.12-288.17), Nipsus describes how a surveyor restores boundaries and boundary stones in an area that was surveyed long ago and has subsequently fallen into disrepair, with the boundary lines obscured and the stones lost. Using the few remaining boundary stones, he attempts to extend the boundary
288:
Nipsus' text never mentions
Pythagoras or Euclid by name. Nipsus' calculations have little practical use and are more practice exercises for scholars. Thus, for two Pythagorean triples Nipsus also calculates how further values might be determined, if the sum of the catheti, the hypotenuse and the
56:("Here begins Marcus Junius Nipsus' second book, well"). Unfortunately, almost nothing is known about this individual. In his work, he gives no information on himself and no other references to him have survived. The text often addresses the reader directly, e.g.
132:(short side) goes in a straight line from the landmark to the surveyor's position. The second cathetus is drawn from the surveyor's position along a path approximately parallel to the riverbank. Halfway along this path, a measuring instrument (
123:
Nipsus explains how to measure the width of a river when the other bank is not accessible (e.g., because a hostile force has occupied it). He seeks out a clear landmark on the other bank, like a tall tree. This forms one corner of a
257:, which form the sides of right-angled triangles). To demonstrate a formula for how one can construct a right-angled triangle using any odd whole number as the length of the shortest cathetus, he naturally uses the 3-4-5 triple:
265:
The given number, 3, is multiplied. That gives 9. From this I subtract one. That gives 8. I divide this in half. That gives 4. That will be the base (of the triangle). To the base I add one. That will be the hypotenuse: 5 feet.
149:
Because of the requirement to form the long path along the river bank and the marking lines, this method is quite time consuming. No attempt is made to apply the mathematically "sophisticated"
67:
The work is only transmitted in a fragmentary state. Scholars differ on exactly which texts should be ascribed to
Nipsius. Some ascribe texts attributed to him to another gromatic writer,
642:
262:
datum numerum, id est III, in se. fit IX. hinc semper tollo assem. fit VIII. huius tollo semper partem dimidiam. fit IV. erit basis. ad basem adicio assem. erit hypotenusa, pedum V
71:, instead, others ascribe additional texts to Nipsius. In general, the opinion of Karl lachmann is accepted, which attributes the following three texts to him:
110:. Other works are only found in more recent manuscripts. Jelle Bouma has edited the first two works with English translation and thorough explanatory notes.
190:("boundary lines"). It is possible to produce plots that are not rectangular. Attached drawings clarify the situation. In the third part (Nipsus,
405:
In citations in this article references to the texts of the
Agrimensores are given by page and line number in the editions of Brian Campbell (
669:
368:
334:
245:. However, the excerpts are fragmentary and marred by many reduplications. While Hero discusses the underlying mathematical theory, like the
64:
286.12) which suggests that it was intended as a practical handbook and it is generally agreed that it was written in the 2nd century AD.
153:, which had been known since the first century AD. The task of measuring the width of a river is also referred to by the surveyor
664:
674:
36:
659:
392:
278:
Hero of
Alexander treated the same calculation of the 3-4-5 triple, but he gives the actual Pythagorean theorem:
622:
Die römische
Feldmesskunst. InterdisziplinĂ€re BeitrĂ€ge zu ihrer Bedeutung fĂŒr die Zivilisationsgeschichte Roms.
305:
Anna
Pikulska (UniversitĂ© de Ćodz), « Les arpenteurs romains et leur formation intellectuelle »,
154:
125:
627:
617:
599:
246:
31:
452:
386:
250:
242:
222:
363:. Amy Shell-Gellasch. Washington, D.C.: Mathematical Association of America. 2007. p. 112.
203:
595:
374:
364:
340:
330:
310:
150:
138:
68:
609:
632:
107:
591:
254:
226:
40:, a compilation of Latin works on land surveying made in the 4th or 5th centuries AD.
653:
574:
410:
186:
288.18-289.17), it is explained how plots of land are formed between different long
230:
194:
289.18-295.15), Nipsus describes the division of land into square plots based on a
579:
Die römischen
Agrimensoren und ihre Stellung in der Geschichte der Feldmesskunst.
495:
Die römischen
Agrimensoren und ihre Stellung in der Geschichte der Feldmesskunst.
143:
91:
378:
344:
195:
162:
27:
606:
Walter de
Gruyter, Berlin/Boston 2014, ISBN 978-3-11-029084-4, pp. 131â148.
208:
182:
lines using the orientation of the stones. In the following part (Nipsius,
358:
324:
624:
Vandenhoeck und
Ruprecht, Göttingen 1992, ISBN 3-525-82480-7, S. 375â397.
129:
249:, Nipsus provides only numerical "recipes". In particular, he uses
594:. "Die Mathematik der Agrimensoren â Quellen und Nachwirkung." in
199:
614:
Nachklassischer Unterricht im Spiegel der gromatischen Schriften.
428:
Nachklassischer Unterricht im Spiegel der gromatischen Schriften.
549:
Friedrich Blume, Karl Lachmann, Adolf August Friedrich Rudorff,
586:
The Roman Land Surveyors: An Introduction to the Agrimensores
213:, the plots of wasteland left over in the surveying process.
482:
Marcus Iunius Nipsus â Fluminis varatio, limitis repositio.
469:
Marcus Iunius Nipsus â Fluminis varatio, limitis repositio.
441:
Marcus Iunius Nipsus â Fluminis varatio, limitis repositio.
360:
Hands on history : a resource for teaching mathematics
202:. He goes into various special cases and also explains the
558:
Marcus Iunius Nypsus â Fluminis varatio, limitis repositio
551:
Gromatici veteres. Die Schriften der römischen Feldmesser.
508:
Die Mathematik der Agrimensoren â Quellen und Nachwirkung.
326:
Pappus of Alexandria and the mathematics of late antiquity
564:
Band 77). Peter Lang, Frankfurt 1993, ISBN 3-631-45588-7.
98:
These works are included in the oldest manuscript of the
60:("when you come to an assigned piece of land", Nipsius
34:
questions. His surviving writings are preserved in the
646:(RE). Vol. X,1, Stuttgart 1918, col. 1069 f.
643:
RealencyclopÀdie der classischen Altertumswissenschaft
329:. Cambridge: Cambridge University Press. p. 23.
233:. The information might be sourced mainly from the
457:AusfĂŒhrliches lateinisch-deutsches Handwörterbuch.
280:
307:Revue internationale des droits de l'Antiquité
161:204.24), without providing a solution, and by
8:
54:incipit Marci Iuni Nipsi liber II feliciter
604:In den Gefilden der römischen Feldmesser.
177:In the first part of this text (Nipsus,
298:
384:
284:aÂČ + ((aÂČ - 1)/2)ÂČ = ((aÂČ - 1)/2 + 1)ÂČ
309:, 3e série, t. 51, 2004, p. 205-216,
7:
620:, Luigi Capogrossi Colognesi, ed.,
562:Studien zur klassischen Philologie.
16:Roman surveyor and technical writer
14:
221:After some basic definitions of
84:("Repositioning of Boundaries")
30:, who also dealt with various
1:
100:Corpus Agrimensorum Romanorum
58:cum in agro assignato veneris
52:is introduced with the words
50:Corpus Agrimensorum Romanorum
37:Corpus Agrimensorum Romanorum
26:) was a second-century Roman
670:2nd-century writers in Latin
553:2 Volumes, Berlin 1848â1852.
90:("Measurement by Feet" or "
691:
534:The Roman land surveyors.
128:which he constructs. One
78:("Measurement of Rivers")
229:, this work focusses on
665:Ancient Roman surveyors
675:Ancient mathematicians
391:: CS1 maint: others (
286:
142:) is placed, and the
126:right-angled triangle
44:Name and transmission
519:Hero of Alexandria,
20:Marcus Junius Nipsus
251:Pythagorean triples
247:Pythagorean theorem
660:2nd-century Romans
453:Karl Ernst Georges
323:Cuomo, S. (2007).
243:Hero of Alexandria
223:Euclidean geometry
628:Johannes Tolkiehn
596:Eberhard Knobloch
426:Ulrich Schindel:
370:978-0-88385-976-6
336:978-0-521-03689-4
173:Limitis Repositio
151:intercept theorem
82:Limitis Repositio
682:
639:
537:
532:O. A. W. Dilke:
530:
524:
517:
511:
506:Menso Folkerts,
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289:area are known.
212:
119:Fluminis Varatio
104:Codex Arcerianus
76:Fluminis Varatio
69:Agennius Urbicus
690:
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253:(sets of three
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28:gromatic writer
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592:Menso Folkerts
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584:O.A.W. Dilke,
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48:A work in the
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618:Okko Behrends
615:
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600:Cosima Möller
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587:
583:
581:Leipzig 1876.
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575:Moritz Cantor
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556:Jelle Bouma.
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411:Karl Lachmann
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569:Bibliography
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557:
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497:pp. 104â107.
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489:
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427:
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277:
274:, p. 300.1â5
271:
269:
238:
234:
231:trigonometry
220:
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108:WolfenbĂŒttel
103:
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35:
32:mathematical
23:
19:
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638:(in German)
207: [
134:ferramentum
654:Categories
633:Iunius 108
521:Geometrica
293:References
239:Geometrica
144:hypotenuse
92:Hypotenuse
471:pp. 59 f.
387:cite book
379:760236992
345:122283241
282:aÂČ+ bÂČ=cÂČ
204:subseciva
196:decumanus
163:Frontinus
544:Editions
217:Podismus
169:14.12).
130:cathetus
88:Podismus
602:, ed.,
588:(1971).
510:p. 140.
484:p. 143.
430:p. 387.
311:on line
235:Metrica
188:limites
630::
536:p. 55.
443:p. 15.
409:) and
377:
367:
343:
333:
198:and a
155:Balbus
102:, the
24:Nypsus
211:]
200:cardo
139:groma
114:Works
640:In:
523:8.1.
393:link
375:OCLC
365:ISBN
341:OCLC
331:ISBN
225:and
22:(or
616:in
560:(=
241:of
237:or
106:in
656::
612:.
598:,
577:.
455::
417:).
415:La
407:Ca
389:}}
385:{{
373:.
339:.
272:La
209:de
192:La
184:La
179:La
167:Ca
159:Ca
136:,
94:")
62:La
635:.
413:(
395:)
381:.
347:.
313:.
165:(
157:(
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