Book of Lemmas

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Propositio I in Liber Assumptorum (1661)
The first page of the Book of Lemmas in The Works of Archimedes (1897).

The book of lemmas (also the book of auxiliary sentences ) is a collection of 15 statements about the geometry of circles . It is attributed to the ancient Greek mathematician Archimedes ; however, its authorship is questionable.

history

The Syrian mathematician Thabit ibn Qurra translated the manuscript from Greek into Arabic (title: K. al-Ma'hūdāt fī usūl al-handasa ) in the 9th century and ascribed it to Archimedes; A commentary by Alī ibn Ahmad al-Nasawī has come down to us from the 10th century . In 1661 the text was translated into Latin by Abraham Ecchellensis and used by Giovanni A. Borelli as Archimedis Liber Assumptorum in his work Apollonii Pergaei Conicorum lib. V, VI, VII issued. The English mathematician Thomas L. Heath in turn created an English version of Liber Assumptorum and included it in his anthology The Works of Archimedes in 1897 under the title Book of Lemmas . This volume was - supplemented u. a. a contribution by the Danish mathematician Johan Ludvig Heiberg about Archimedes 'methods - translated into German by Fritz Kliem in 1914 ( Archimedes' works ); the chapter Book of Lemmas is called the Book of Suffices here .

The 15 statements are called in the Liber Assumptorum and in the Book of Lemmas "Propositionen", in the German translation Kliem's ​​" sentences ". The work has not survived in Greek.

Authorship

The authorship of Archimedes is not assured. In particular passages in the text that refer to Archimedes in the third person raise doubts. In sentence 4, for example, there is talk of a figure (meaning the Arbelos ) who is called “a so-called Άρβυλος by Archimedes” (“quam vocat Archimedes ARBELON” or “what Archimedes called an Άρβυλος”).

On the question of Archimedes' authorship, Heath explains (Kliem's ​​translation to the right):

“The Lemmas cannot, however, have been written by Archimedes in their present form, because his name is quoted in them more than once. The probability is that they were propositions collected by some Greek writer of a later date for the purpose of elucidating some ancient work, though it is quite likely that some of the propositions were of Archimedean origin, eg those concerning the geometrical figures called respectively Άρβυλος ( literally 'shoemaker's knife') and σαλινον (probably a 'salt-cellar'), and Prop. 8 which bears on the problem of trisecting an angle. "

“However, the auxiliary sentences cannot be written by Archimedes in the current form, as his name is mentioned several times in it. Probably they were sentences collected by a later Greek writer to explain an ancient work, but it is very likely that some of the sentences are of Archimedean origin, e.g. B. those who refer to the geometric figures with the names Άρβυλος (literally "cobbler knife") and σαλινον (perhaps "salt barrel"), and sentence 8, which deals with the problem of the three-dimensional division of the angle. "

In summary, this means that at least the Arbelos (the above-mentioned “shoemaker's knife”), the Salinon (the “salt barrel”) and the method of dividing the angle into three parts can be attributed to Archimedes with high probability.

content

The 15 sections of the text contain statements about circles, their diameters and radii , secants and tangents and the relationships between these elements, as well as the associated evidence . They are all illustrated in the Latin version, with the exception of sentence 7 in the English and German versions.

Sentence 2 may serve as an example , quoted and illustrated by Kliem:

Book of Lemmas Prop 2.png

“ Let AB be the diameter of a semicircle and the tangents to it in B and any other point D may intersect in T. If the perpendicular DE falls from D to AB and AT and DE intersect in F, then DF = FE.
We extend AD up to the intersection H with the extension of BT. Then the angle ADB in the semicircle is a right one; consequently the angle BDH is also a right one. TB, TD are the same. Hence T is the center of the semicircle over BH as the diameter that goes through D; thus HT = TB. Since DE and HB are now parallel, it follows that DF = FE. "

Arbelos and Salinon

In particular, the two more complex geometric figures Arbelos and Salinon , each consisting of several semicircles, are introduced: The Arbelos himself in sentence 4, the twin circles of Archimedes in sentence 5, the inscribed circle of Arbelos (which in turn is related to the Pappos chain , such as Kliem notes in a footnote) in sentence 6, and finally the Salinon in sentence 14.

  • Arbelos

  • Twin circles

  • Pappos chain

  • Salinon

Individual evidence

  1. ↑ a b Liber assumptorum. In: Infothek der Scholastic. University of Regensburg, accessed April 20, 2012 .
  2. ↑ From Euclid to Newton: An Exhibition in Honor of the 1999 Conference of the Mathematical Association of America. Brown University Library, accessed May 15, 2016 .
  3. ^ Aaboe: Episodes from the Early History of Mathematics. 1998, p. 77
  4. ^ Kliem: Archimedes' works. 1914, p. VII
  5. ^ Kliem: Archimedes' works. 1914, p. 456 ff
  6. ^ Kliem: Archimedes' works. 1914, p. 459
  7. ↑ Borelli: Apollonii Pergaei Conicorum ... 1661, p. 390
  8. ^ Heath: The works of Archimedes. 1897, p. 304
  9. ^ Heath: The works of Archimedes. 1897, p. Xxxii
  10. ^ Kliem: Archimedes' works. 1914, p. 21.
  11. ^ Kliem: Archimedes' works. 1914, p. 457
  12. ^ Kliem: Archimedes' works. 1914, p. 462

literature

  • Giovanni A. Borelli et al .: Apollonii Pergaei Conicorum Lib. V, VI, VII & Archimedis Assumptorum Liber . ex typographia Iosephi Cocchini ..., Florence 1661, p. 379-413 .
  • Thomas L. Heath : The works of Archimedes . University of Cambridge, Cambridge 1897, p. xxxii, 301-318 .
  • Fritz Kliem : Archimedes' works: With modern designations / ed. u. with e. Inlet provided by Sir Thomas L. Heath. German by Fritz Kliem . O. Häring, Berlin 1914, p. 456-470 .
  • Asger Aaboe : Episodes from the Early History of Mathematics . tape 13 . The Mathematical Association of America, Washington, DC 1998, ISBN 0-88385-613-1 , pp. 77 .

Web links

Digital copies

Facsimiles of the Latin ( Liber Assumptorum ), the English ( Book of Lemmas ) and the German ( Buch der Hilfssätze ) versions are available as digital copies, both for online reading and for downloading as PDF documents.

Visualizations