The elements (in the original Στοιχεῖα Stoicheia ) are a treatise by the Greek mathematician Euclid (3rd century BC), in which he summarizes and systematizes the arithmetic and geometry of his time. For the first time, the work exemplifies the structure of an exact science, since most statements are derived and proven from a limited store of definitions , postulates and axioms . To this day, this approach has not only influenced mathematicians, but also many physicists, philosophers and theologians in their attempt to base their science on axioms.
The individual books
Euclid's Elements originally consisted of 13 books with the following topics in the individual books (the presumed sources in brackets):
- Book 1: From Definitions to the Pythagorean Theorem ( Pythagorean )
- Book 2: Geometric Algebra (Pythagoreans)
- Book 3: Circle theory (Pythagoreans)
- Book 4: Polygons (Pythagoreans)
- Book 5: Irrational Quantities ( Eudoxus of Knidos )
- Book 6: Proportions (sources unknown)
- Book 7: Divisibility and Prime Numbers , e.g. B. the lemma of Euclid (Pythagorean)
- Book 8: Squares , Cubes, and Geometric Series (Pythagoreans)
- Book 9: Even and Odd Numbers , etc. a. also the Euclid (Pythagorean) theorem
Book 11–13: Spatial Geometry
- Book 11: Elementary things about spatial geometry
- Book 12: Exhaustion Method (Eudoxos by Knidos)
- Book 13: The Five Even Bodies (Theaetetus)
Two more were later added to these books, but they are not counted among the actual elements:
- Book 14: A book by Hypsicles (2nd century BC) on icosahedra and dodecahedron.
- Book 15: A book by a later unknown author, preserved only in Arabic and Hebrew, preserved in Arabic in commentaries from Muhyi al-Dīn al-Maghribī . It deals with the five regular polyhedra. Possibly it goes back to Greek tradition.
Euclid's elements contain many important mathematical results. In part, the elements are the oldest source for these results. The following are known, for example:
- Book 1, Postulate 5: Axiom of Parallels
- Book 1, Proposition 43: Theorem of the Gnomon
- Book 1, Proposition 47: Pythagorean Theorem
- Book 1, Proposition 32: Sum of angles in a triangle
- Book 2, Proposition 4: First Binomial Formula
- Book 3, Proposition 20: Circle Angle Theorem
- Book 3, Proposition 31: Theorem of Thales
- Book 4, Proposition 11: Construction of the Regular Pentagon
- Book 5, Proposition 25: Special case of the inequality of the arithmetic and geometric mean
- Book 6, Proposition 2: First Ray Theorem
- Book 6, Proposition 4: Similarity Theorems for the Triangle
- Book 7, Proposition 2: Euclidean Algorithm
- Book 9, Proposition 20: Existence of Infinitely Many Prime Numbers (Euclid's Theorem)
- Book 10, Proposition 10: Irrationality of the Square Root of 2
- Book 12, Proposition 10: Determination of the cone volume
- Book 13, Proposition 8: Golden ratio in the regular pentagon
- Book 13, Proposition 18 a: Classification of the Platonic Solids
The oldest surviving Greek manuscript comes from Byzantium of the year 888 and is today kept in the Bodleian Library ( Oxford ) (MS. D'Orville 301) and corresponds to the edition in the arrangement by Theon of Alexandria. Of particular importance is a Greek manuscript from the 10th century in the Vatican library (Vaticanus Graecus 190, named by Heiberg P ), which contains a text before the editing by Theon and on which all recent editions are based. It contains books 1 to 15 (as well as the data, Marinos' commentary on the data and some texts by Theon). There is also a papyrus fragment found in Oxyrhynchus in 1897 (Oxyrhynchus 29, University of Pennsylvania Library), probably from the period AD 75 to 125, but containing only Proposition 5 from Book 2, and a papyrus fragment from Herculaneum (No. . 1061) with definition 15 from Book 1.
Both the version by Theon and Vatican 190 contain explanations or minor additions by older editors, the so-called Scholia.
Of the numerous Arabic translations and commentaries, the two only fragmentarily known translations of al-Haggag (or al-Hajjaj) from the end of the 8th century and those of Ishaq ibn Hunayn / Thabit ibn Qurra (end of the 9th century) were particularly important for tradition. or by Nasir Al-din al-Tusi (1248) of importance. Al-Nayrizi wrote a commentary in the early 10th century based on the translation by Al-Haggag.
We owe the first medieval translation of the elements into Latin to the Englishman Adelard von Bath . He roamed Europe in the 12th century in search of manuscripts and around 1120 transferred this work from Arabic. Separately, the elements were translated from Arabic in the same century in Spain by at least two other famous translators: Hermann of Carinthia and Gerhard of Cremona . The most influential of the early translations was that of Campanus de Novara (c. 1260) into Latin (he used the translation by Adelhard von Bath and other works), which was also later the basis of the first prints of the elements and was the dominant until the 16th century The issue was when translations directly from Greek replaced them. All in all, the elements are one of the most common texts handed down in manuscripts from the Middle Ages in numerous different, mostly fairly free versions.
Also in the 12th century, another translation of the elements from Greek was made in southern Italy or Sicily , but the author is unknown. Because of the style of translation it seems likely that it is in this author by the same, which in 1160 also the Almagest of Ptolemy translated. She was identified by John E. Murdoch in 1966 .
Of course, The Elements were one of the first works to be printed. The first Latin edition, based on the translation by Campanus of Novara, appeared in Venice in 1482. The preparatory work on the Regiomontanus remained unfinished in the 1460s. A complete translation from the Greek by Bartolomeo Zamberti (or Zamberto, 1473 to after 1543) could then be printed in 1505. From this period after the invention of printing , only a few important works are highlighted here: the translation of Federicus Commandinus (1509–1575) from Greek (1572), the extensively commented edition by Christoph Clavius (1574) and the Oxford edition of Works of Euclid by David Gregory 1703 (Greek / Latin). All of these were translations into Latin. The first translation into a modern Western language was published by Niccolò Tartaglia in 1543. The first German translation appeared in 1555 ( Johann Scheubel ) and Simon Marius published the first German translation (of the first six books) directly from Greek in 1609. The first English translation appeared in 1570 ( Henry Billingsley ), the first French in 1564 ( Pierre Forcadel ), the first Spanish in 1576 ( Rodrigo Zamorano ) and the first in Dutch in 1606 ( Jan Pieterszoon Dou ). The first translation into Chinese (the first parts) was done by Xu Guangqi and Matteo Ricci (1607). In the popular editions from the 16th century onwards, however, simplifications and revisions were often made, the evidence replaced by examples, and in most cases not all books of the elements were published, but typically only the first six.
The first printed edition of the Greek text (Editio Princeps) was published by Simon Grynaeus in Basel in 1533.
Well-known commentaries were written by Proclus and Theon of Alexandria in antiquity , who also made his own arrangements in his edition of the elements. All editions known in the West were based on text versions that came from Theon and his school until the 19th century. Only Johan Ludvig Heiberg reconstructed an original version based on a manuscript (Vaticanus Graecus 190) that was not based on the tradition of Theon, and which François Peyrard discovered among the books confiscated by Napoleon from the Vatican. From this, Peyrard published a French translation in 1804 and 1814-1818, and it was used in Germany by Johann Wilhelm Camerer and Karl Friedrich Hauber for their Latin-Greek edition of the first six books in 1824/25. After the Heiberg edition, Clemens Thaer published a German translation (1933–1937).
In 1996, Wilbur Richard Knorr criticized Heiberg's text edition, which follows the Greek lineage and is considered the standard edition, based on the Arabic lineage (which Heiberg had dealt with in 1884 on the occasion of Martin Klamroth's research on the Arabic Euclid ).
- Euclidean geometry
- Axiom of parallels
- Euclidean Algorithm
- Euclid's proof of the irrationality of the root of 2
- Euclid's Elements, fifteen books , translated by Johann Friedrich Lorenz, Halle 1781 ( online );
- Johan Ludvig Heiberg, Heinrich Menge (editor): Euclidis Opera Omnia. Leipzig, Teubner, from 1888, volumes 1–5 with the elements (including the Scholia in volume 5), re-edited by Teubner from 1969 to 1977 as Euclidis Elementa (Greek, Latin).
- Euclid: The elements. Books I – XIII. Ed. U. trans. v. Clemens Thaer . Frankfurt a. M. Harri Deutsch, 4th edition 2003 (= Ostwalds Klass. D. Exacten Wiss. 235, 236, 240, 241, 243). The translation first appeared in 1933–1937. ISBN 3-8171-3413-4 .
- Max Steck : Bibliographia Euclideana. The spiritual lines of tradition in the editions of the "Elements" of Euclid (around 365-300). Manuscripts, incunabula, early prints from the 16th century. Critical editions of the 17th – 20th centuries Century. Editions of the Opera Minora (16th – 20th centuries). Reprint, ed. v. Menso Folkerts . Hildesheim: Gerstenberg, 1981.
- Benno Artmann : Euclid - The Creation of Mathematics. New York, Berlin, Heidelberg: Springer 1999, ISBN 0-387-98423-2 (English introduction to the structure and proof of the elements ).
- The thirteen books of Euclid's elements. Ed. U. trans. v. Thomas Heath , 3 volumes, Cambridge University Press 1908, reprint Dover 1956 (English translation with extensive commentary and introduction to Euclid).
- Max Simon : Euclid and the six planimetric books. Leipzig. Teubner 1901 (translation of the first six books).
- Martin Klamroth : About the Arabic Euclid , Journal of the German Oriental Society, Volume 35, 1881, pp. 270–326, online .
- John Emery Murdoch : Euclid: Transmission of the Elements , Dictionary of Scientific Biography , Volume 4, 1971, pp. 437-459.
- The elements of Euclid, Euclides: Stoicheia. The elements translated into German.
- Norbert Froese, Euclid and the Elements. Structure and content of the elements (37 pages, PDF document; 844 kB).
- WD Geyer: Euclid: The elements - an overview. Lecture on ancient mathematics SS 2001 (PDF, 275 kB). At Uni-Bielefeld.de.
- Euclid's Elements. Online version with Java applets and bibliography by DE Joyce.
- Euclid's elements. The original Greek text.
- The importance of Euclid and the elements is proven by the mathematician and philosopher Paul Lorenzen , who writes in the introduction to his elementary geometry (p. 9): Euclid is the most famous geometer in the world. The Bible and Euclid's "Elements" were by far the most widely read books for nearly 2,000 years. What we mean by "math" in school jargon is still called "Euclid" in the English-speaking world. But while Old Testament readers quarreled over New Testament and Quran readers, Euclid retained the same interest among Jewish, Christian, and Islamic scholars. Although almost nothing is known of his life (around 300 ante in Alexandria at the court of the first Ptolemy), there have been Latin translations since late antiquity and Arabic translations since around 800. In scholasticism, the translation by Campanus (13th century based on an Arabic model) was decisive - it was printed in 1482. The elements were used as school books well into the 19th century.
- The geometer and mathematician Howard Whitley Eves expresses himself in the same way in An Introduction to the History of Mathematics (p. 115): … No work, except the Bible, has been more wideley used, edited or studied, and probably no work has exercised a greater influence on scientific thinking. Over a thousand editions of Euclid's Elements have appeared since the first one printed in 1482, and more than two millennia this work has dominated all teaching of geometry.
- by Maximilian Curtze in 1899 as a supplement to the Heiberg edition in the Latin translation by Gerhard von Cremona .
- Hubert LL Busard : The first Latin translation of Euclid's elements commonly ascribed to Adelhard of Bath. Toronto 1983. There are three different versions of Adelhard's translation ( Marshall Clagett : The medieval latin translation from the Arabic of the Elements of Euclid with special emphasis on the versions of Adelhard of Bath. Isis, Volume 44, 1953, pp. 16– 42).
- Hubert LL Busard: The latin translation of the arabic version of Euclids Elements commonly ascribed to Gerard of Cremona. Leiden: Brill 1984.
- John Murdoch: Euclides Graeco-Latinus: a hitherto unknown medieval latin translation of the elements made directly from the Greek. Harvard Studies in Classical Philology, Volume 61, 1966, pp. 249-302.
- This translation was edited by Hubert LL Busard: The medieval latin translation of Euclid's elements made directly from the Greek. Stuttgart: Steiner 1987.
- Scheubel: The sibend, eight and ninth book of the highly famous Mathematici Euclidis Megarensis , Augsburg 1555, after Ulrich Reich Scheubel, Johannas. Neue Deutsche Biographie 2005. According to Thomas Heath, the first German translation in Basel in 1558 was the translation of Books 7 to 9 by Johann Scheubel, who had already published a Greek-Latin edition of the first six books in 1550. Xylander published a German edition of the first six books in 1562, but it was intended for practitioners and took great liberties (omitting the evidence, among other things).
- Complete translation. Sir Henry Billingsley (died 1606) was a wealthy London merchant who became Lord Mayor of London in 1596.
- A pupil and friend of Pierre de la Ramée and professor of mathematics at the Collège Royale. He translated nine books just like Errard de Bar-le-Duc in the second French edition in 1604.
- The first six books.
- Scriba, Schreiber: 5000 years of geometry. Springer Verlag 2005, p. 249.
- He himself made a reference to this in his commentary on the Almagest, where he wrote the last proposition in Book 6 of his version that the element came from him. Peyrard noticed in the Vatican manuscript that this proposition was missing.
- Knorr, The wrong text of Euclid: On Heiberg's text and its alternatives, Centaurus, Volume 38, 1996, pp. 208-276