Geometria (Gerbert d'Aurillac)

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Geometria (Gerbert d'Aurillac) is an extensive collection of geometric , but also metrological subjects that have been handed down through ancient manuscripts. It was written in Middle Latin in the second half of the 10th century and is wholly or partly ascribed to Gerbert d'Aurillac .

Sources used

Gerbert d'Aurillac had an extensive library available during his time as Abbot of Bobbio . Numerous ancient writers, from Plato ( Timaeus in the translation of Calcidius ) to the church father Augustine and the late antique Martianus Capella were able to give him suggestions. In particular, however, he used a collection with the writings of the Roman Agrimensors ( Corpus agrimensorum Romanorum ), which was also there. But the author also used sources closer to him in terms of time. This is supported by the use of the abacus and astrolabe , which were not known in antiquity, and the similarity in content to the collection of exercises Propositiones ad acuendos iuvenes by the early medieval scholar Alkuin .

Structure and content

Since the Geometria in medieval manuscripts was often compiled with fonts of similar content by other authors, there was considerable controversy about the scope of the font. While Jacques Paul Migne recorded the entire text in XCIV in the volume CXXXIX of the Patrologia Latina , this was not the case with later editions. It is undisputed that the work is divided into the following 3 very different parts:

  • Chapters I-XIII: geometrical definitions and elementary theorems, use of the abacus
  • Chapter XIV-XL: practical geometry with height and distance measurements
  • Chapter XLI-XCIV: computational geometry with exercises

The opinion has prevailed that only Chapters I-XIII can definitely be assigned to Gerbert d'Aurillac. Chapters XIV-XCIV were outsourced by Nicolaus Bubnov under the title Geometria incerti auctoris ( Geometry of an uncertain author ).

Geometrical definitions and elementary sentences, use of the abacus

After the prologus , in which he praises the importance of the Egyptians for geometry following his ancient models, the author gives some definitions from Book I of the elements of Euclid , from point to solidus ( body ). In doing so, he refers to Boethius . He could find excerpts from his writings in the Corpus agrimensorum Romanorum .

In Chapters II and III he lists the length and area of ​​the antiqui , from the smallest unit, the digitus, to the iugerum and the stadium . Balbus (Agrimensor) gave similar information , but also Isidore of Seville ( Etymologiae , XV, XV).

Then he continues with the Euclidean definitions: straight-line figures, right-acute-obtuse angles, parallels, etc. In order to then carry out essential sentences about triangles from this book in Chapters V to XI. This ends §48 with a presentation of the Pythagorean theorem .

The following chapters contain arithmetic problems. In a right-angled or equilateral triangle, 2 values ​​are given, others can be calculated. In antiquity, these tasks, which were also popular with earlier authors, could only be carried out with selected numerical examples, in which the entire calculation remained in the integer range. Gerbert d'Aurillac is not afraid of rational numbers . With the help of the abacus he can perform difficult arithmetic operations. Ut abacistae facillimum est, in se ducens (as is very easy for the abacist, the number is multiplied by itself) , he writes. And so he solves a problem in which one of the given values ​​leaves the integer with 6 1/3 feet (Chapter XII).

Practical geometry

In chapters XIV to XL the practical execution of survey work is described, the determination of river widths, tower heights, well depths. These tasks are in the tradition of the Roman agrimensors, but are not a direct takeover from the Corpus Agrimensorum Romanorum . Chapters XXXVIII and XXXIX describe the calculation of a river width like the fluminis variatio of the Agrimensor Marcus Iunius Nipsus , but with a different construction approach. In addition, the astrolabe , unknown to the agrimensors, is used as an aid (Chapter XVI and others).

Computing geometry with exercises

These chapters, on the other hand, are more closely committed to the Corpus agrimensorum Romanorum . They are arithmetic tasks that are far from practical and involve right-angled triangles, circular areas, various rectangles. Skilful selection of numbers ensures that the space of whole numbers is not left. Many examples are taken from the Agrimensors Marcus Junius Nipsus and Epaphroditus . Even mistakes that have crept in are passed on, such as a wrong expression taken over from Marcus Junius Nipsus in Chapter XLII or a number spelling in Chapter L (24 * 7 = 169, like Epaphroditus, § 11). On the other hand, chapters LXVII to LXXI correlate strongly with the Propositiones ad acuendos iuvenes 21-25 and 27-31 of Alcuin. The tasks that appear to have been taken from everyday life (Chapter LXVII: how many sheep fit in a certain rectangular field? Chapter LXXVI: how many wine barrels in a cellar?) Are correct in the numerical examples and even in the formulations match. It remains to be seen whether Gerbert d'Aurillac knew the somewhat older script or whether both used a now lost source.

The letter to Adalbald II

The written exchange between the author and the scientifically active Adelboldus Traiectensis (Adalbald II, Bishop of Utrecht) is closely linked to Geometria . Adalbald dedicated his geometrical / cosmological work De ratione inveniende crassitudinem sphaerae to Gerbert d'Aurillac and apparently asked him several times for advice on geometrical questions. In a letter written between 983 and 997, Gerbert d'Aurillac not only dealt with his question competently, but also explains the problem using a numerical example and a drawing.

Tradition and survival

The work has come down to us in numerous manuscripts as part of collections of relevant texts. These bundles consist of ancient books, such as the Timaeus translation by Calcidius , mathematical works such as the geometry ascribed to Boethius , and pieces from the Corpus agrimensorum Romanorum . Gerbert d'Aurillac made an important contribution to the development of geometry in the scholastic school system, especially to the Practica geometriae of Hugo von St. Viktor .

The Benedictine Father Bernhard Pez edited the work in 1721. A. Olleris published a publication in 1867 as part of the complete work. Nicolaus Bubnov dealt extensively with various aspects of the text in his book. A translation into the German language is not available.

Text output

  • Nicolaus Bubnov: Gerberti Opera Mathematica (972-1003) , Berlin 1899

literature

  • Moritz Cantor : The Roman Agrimensors and their position in the history of field measurement art , Leipzig 1875
  • Menso Folkerts : Boethius Geometry II, Wiesbaden 1970
  • Menso Folkerts: The oldest collection of mathematical exercises in Latin: The PROPOSITIONES AD ACUENDOS IUVENES ascribed to Alkuin. , Vienna 1978
  • Menso Folkerts: The Mathematics of Agrimensors - Sources and Aftermath in In the Fields of the Roman Surveyors (Eds. Eberhard Knobloch, Cosima Möller), Berlin / Boston 2014
  • Frederick A. Homann: PRACTICAL GEOMETRY, [Practica Geometriae] Attributed to Hugh of St. Victor , Milwaukee 1991
  • Uta Lindgren : Gerbert von Aurillac and the Quadrivium , Wiesbaden 1976
  • Kurt Vogel : Gerbert von Aurillac as mathematician in Acta historica Leopoldina No. 16, Halle 1985

Individual evidence

  1. ↑ Menso Folkerts: The Mathematics of Agrimensors - Sources and Aftermath , p. 145
  2. ↑ Uta Lindgen: Gerbert von Aurillac and the Quadrivium , p. 24
  3. ↑ Moritz Cantor: The Roman Agrimensors and their position in the history of field measurement art , p. 160f
  4. ↑ Uta Lindgren: Gerbert von Aurillac and the Quadrivium , p. 24
  5. ↑ Kurt Vogel: Gerbert von Aurillac as a mathematician , p. 19
  6. ↑ Nicolaus Bubnov: Gerberti Opera Mathematica (972-1003) , page 58, Note 1
  7. ↑ Nicolaus Bubnov: Gerberti Opera Mathematica (972-1003) , S. 84f
  8. ↑ Kurt Vogel: Gerbert von Aurillac as a mathematician , p. 20
  9. ↑ Menso Folkerts: The Mathematics of Agrimensors - Sources and Aftermath , p. 145
  10. ↑ Nicolaus Bubnov: Gerberti Opera Mathematica (972-1003) , Geometria incerti auctoris , pp 336-364, comments
  11. ↑ Nicolaus Bubnov: Gerberti Opera Mathematica (972-1003) , p 339, note 38
  12. ↑ Menso Folkerts: The oldest collection of mathematical exercises in Latin , p. 40
  13. ↑ Patrologia Latina CXL, 1103
  14. ↑ Nicolaus Bubnov: Gerberti Opera Mathematica (972-1003) , p 487
  15. ↑ Kurt Vogel: Gerbert von Aurillac as a mathematician , p. 21
  16. ^ Menso Folkerts: "Boethius" Geometry II , pp. 3-39
  17. ↑ Frederick A. Homann: PRACTICAL GEOMETRY, [Practica Geometriae] Attributed to Hugh of St. Victor , pp. 5f
  18. ↑ Moritz Cantor: The Roman Agrimensors and their position in the history of field measurement art , p. 152
  19. ↑ Moritz Cantor: The Roman Agrimensors and their position in the history of field measurement art , p. 150, note 276