Epaphroditus (Agrimensor)

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Epaphroditus is the name of a Latin specialist writer, an agrimensor (surveyor), who is associated with an extensive mathematical text in the Corpus agrimensorum Romanorum . (Since the texts of the Agrimensors are poorly structured, in the following the quotations of Epaphroditus are indicated by the paragraph in the edition used, those of the other Agrimensors by the name of the Agrimensor, the abbreviation of the editor ( Ca for Brian Campbell) and the page and Line number specified.)

Name, person, tradition

The Corpus agrimensorum Romanorum contains the most extensive mathematical treatise of antiquity in Latin. In addition to an extensive collection of definitions, it contains a description of mathematical relationships that go beyond anything else preserved in the Latin language. The text is in the manuscript Codex Arcerianus from the 6th or 7th century, which is kept in Wolfenbüttel, between the Agrimensors Marcus Iunius Nipsus and Sextus Iulius Frontinus . There are only a few editions, including those by Nikolaj M. Bubnov (1899) and Moritz Cantor . A German translation is not available.

The chronological classification extends from the 2nd century AD to the middle of the 4th century AD. The text begins and ends by naming its two authors Aprofiditus and Betrubus Rufus Architect or Aprofoditus and Bertrubus Rufus Architect . The first name probably derives corrupted from Epaphroditus. Betrubus sounds like Marcus Vitruvius Pollio , but it is very likely that it has nothing to do with the Roman architect. There are no testimonials about either author, the text does not contain any self-testimonials.

Content, sources

The text was divided into 40 paragraphs by the editor Moritz Cantor. The following presentation is based on the context of the content and not on the order of the paragraphs.

geometry

A large part of the script (especially §§ 1–6, §§ 10–14) belongs to geometry with non-practical “school tasks”. The source is largely Heron of Alexandria . But also with Marcus Junius Nipsus there are similar tasks for the calculation of right-angled triangles, whereby the Pytagorean triples are used in particular . However, Section 30 goes beyond that. This formula about the inscribed circle can otherwise only be found in the ancient Latin-speaking area with the late ancient Roman scholar Anicius Manlius Severinus Boethius . Without the usual and cumbersome cladding in a numerical example, Epaphroditus writes:

“ In trigonum hortogonium circulum inscribere, qui omnes eius lineas tangat. sic. iunge chatetum et vasem, deme ypotenusam. erit diametron circuli "

"Calculation of the inner circle diameter for a right triangle: add the legs and subtract the hypotenuse , the diameter of the circle will be."

Polygonal numbers, pyramidal numbers

In § 15, §§ 17-24, Epaphroditus deals with calculating the area of regular polygons , the side length of which is given, from 3-corner to 12-corner. At least that's what he writes. In fact, it does not calculate the area, but the polygonal number , i.e. for an m-gon with side length r the sum of an arithmetic sequence of r terms, the starting term of which is 1 and the difference m-2. Or the sum of the units, which can be used to represent the m-corner, clearly, even if inexactly expressed. The number determined in this way is an approximation of the area, albeit somewhat larger (except for the square numbers ) and can be referred to as "arithmetic area determination". The text always first gives the formula in a cumbersome way, and then calculates a numerical example:

" ... eptagonus ... cuius latus unum in se multiplico, et postea quinquies duco. ipsa aera ter duco. diamidiam partem sumo eptagonum dico "

“... heptagon ... I multiply a page with itself and then with five. I subtract the number given at the beginning three times. I take half. I call this heptagon ... or ... heptagon number = (5r² - 3r) / 2 "

In addition, the pyramidal numbers are also given.

The polygonal numbers are an essential element of Pythogarean number theory and were common throughout ancient Greece. They entered the Latin technical literature through the De institutione arithmetica des Boethius. However, only the material for the development of the formulas of Epaphroditus can be found there. Because Boethius writes of the plana superficies in numeris and gives the first links for 3-sided to 7-sided and also the summation of these arithmetic sequences. However, he does not use any formula for calculating these sums from his Greek sources.

Practical representations and definitions

Some paragraphs deal with problems that have a certain relation to the activity of the agrimensor and also of arable farming. These are:

  • § 7. Calculation of the number of trees for evenly planting a rectangular plot of land. Lucius Junius Moderatus Columella writes this similarly.
  • § 8, § 9. Approximate calculation of the area of ​​a hill by averaging over several measurable distances.
  • § 25-29. Calculation of the area of ​​the circle from the radius using the ancient approximation pi = 22/7.
  • § 31, 32. Length and area measurements from the centuria to the digitus and their conversion ratios. This is found in a very similar way with Balbus (Agrimensor) . As with Balbus, the definition of the three angles rectus, acutus, hebes = "right, acute, obtuse" follows . However, the explanations are much more limited.

Text output

  • Nicolaus Bubnov: Gerberti Opera Mathematica , ext . VII, 5, 6, Berlin 1899
  • Moritz Cantor : The Roman Agrimensors and their position in the history of field measurement art. Teubner, Leipzig 1875, note 230 on pp. 207–215 ( digitized version ).

literature

Single receipts

  1. ↑ Menso Folkerts: The Mathematics of Agrimensors - Sources and Aftermath , p. 132
  2. ^ Moritz Cantor: The Roman Agrimensors and their position in the history of field measurement art. Pp. 122, 129
  3. ^ Friedrich Hultsch: Epaphroditos 6. In: Paulys Realencyclopädie der classischen Altertumswwissenschaft (RE). Volume V, 2, Stuttgart 1905, Col. 2714 f.
  4. ↑ Nikolai M. Bubnov: Gerberti postea Silvestri II papae Opera Mathematica. Friedländer, Berlin 1899, pp. 517–551 ( digitized version )
  5. ↑ Menso Folkerts: The mathematics of the agrimensors - sources and aftermath. P. 132
  6. ^ Moritz Cantor: The Roman Agrimensors and their position in the history of field measurement art. , Pp. 115-118
  7. ^ Moritz Cantor: The Roman Agrimensors and their position in the history of field measurement art. P. 118
  8. ↑ Marcus Junius Nipsus: Podismus , Ca , p 297-305
  9. ↑ Boethius: Ars geometrica , II, Item de eodem
  10. ^ Moritz Cantor: The Roman Agrimensors and their position in the history of field measurement art. P. 121
  11. ↑ Menso Folkerts: The mathematics of the agrimensors - sources and aftermath. P. 138
  12. ↑ Boethius: De institutione arithmetica , II, 6-14
  13. ↑ Columella: De re rustica , Book 5.3
  14. ↑ Menso Folkerts: The mathematics of the agrimensors - sources and aftermath. , P. 137.
  15. ↑ Balbus: Expositio et ratio omnium formarum , Ca , p. 206