Euclid's Elements

Euclid's elements or the elements ( in the original Στοιχεῖα STOICHEIA ) is a treatise by the Greek mathematician Euclid (3rd century BC), in which he summarizes and systematizes the arithmetic and geometry of his time. This work shows for the first time pattern of the structure of an exact science, as most statements are derived from a limited supply of definitions, postulates and axioms, and proved. This approach influenced to this day not only mathematicians but also many physicists, philosophers and theologians in their attempts to build their science on axioms.

The elements have been used for 2,000 years as an academic textbook and were up in the second half of the 19th century the most widely according to the Bible work of world literature.

The individual books

Euclid's Elements originally consisted of 13 books on the following topics in the individual books ( in brackets are the suspected sources):

Book 1-6: surface geometry, including congruent and similar figures

• Book 1: From the definitions to the theorem of Pythagoras ( Pythagorean )
• Book 2: Geometric Algebra ( Pythagoreans )
• Book 3: District Administration ( Pythagoreans )
• Book 4: polygons ( Pythagoreans )
• Book 5: Surds ( Eudoxus of Cnidus )
• Book 6: Proportions ( source unknown)

Book 7-9: Arithmetic, including number theory and theory of proportions

• Book 7: divisibility and prime numbers, for example, the lemma of Euclid ( Pythagoreans )
• Book 8: square, cubic number and geometric series ( Pythagoreans )
• Book 9: Even and odd numbers, including also the set of Euclidean ( Pythagorean )

Book 10: Geometry for incommensurable magnitudes ( Theaetetus )

Book 11-13: space geometry

• Book 11: Elemental for room geometry
• Book 12: exhaustion ( Eudoxus of Cnidus )
• Book 13: The five uniform body ( Theaetetus )

These books were later added two more:

• Book 14: A book of Hypsicles ( 2nd century BC)
• Book 15: A book probably of Damascius (5th century AD)

Tradition

The oldest surviving Greek manuscript dates from the Byzantine Empire of the year 888, and is now in the Bodleian Library ( Oxford) preserved ( MS. D' Orville 301) and corresponds to the output in the processing of Theon of Alexandria. Of particular importance is a Greek manuscript from the 10th century in the Vatican Library ( Vaticanus Graecus 190, called by Heiberg P), which contains a text from being edited by Theon and are based on the all new editions. It contains book 1 to 15 ( as well as the data, the comment from Marinos to the data and some texts of Theon ). There is also a 1897 found in Oxyrhynchus Papyrus Fragment ( Oxyrhynchus 29, Library of the University of Pennsylvania ), probably from the time of 75 to 125 AD, but it only contains the Proposition 5 of Book 2, and a papyrus fragment from Herculaneum ( no. 1 061 ) with the definition of 15 from book 1

Both the version of Theon and Vatican 190 contain explanations or smaller additions of older agents, the so-called Scholia.

A translation of Boethius from the Greek into Latin ( by 500 ) is now obtained only partially and only in later edits. Earlier translations into Latin (before Boethius ) are lost.

Of the numerous Arabic translations and commentaries were for the tradition especially the two only fragmentarily known translations of al - Hajjaj (or al - Hajjaj ) towards the end of the eighth century and those of Ishaq ibn Hunayn / Thabit ibn Qurra (late 9th century ) or by Nasir al- Din al - Tusi ( 1248 ) are of importance. Al- Nayrizi wrote at the beginning of the 10th century a comment, based on the translation of Al- Hajjaj.

The first medieval translation of the Elements into Latin we owe the Englishman Adelard of Bath. This roamed Europe in the 12th century in search of manuscripts and rendered so around 1120, this work from Arabic. Regardless of the elements in the same century were transferred to Spain from at least two other famous translators from the Arabic by Hermann of Carinthia and Gerard of Cremona. The most influential of the early translations was that of Campanus of Novara ( 1260 ) into Latin ( he used the translation of Adelhard of Bath and other works ), which was later the basis of the first prints of the elements and to the 16th century the dominant issue was who succeeded as translations directly from the Greek this. Overall, the elements are one of the most traditional in manuscripts texts of the Middle Ages in many different, often quite free versions.

Also in the 12th century originated in southern Italy or Sicily another translation of the elements from the Greek, whose author is unknown. Because of the style of the translation, it seems likely that it is this author is the same, who also translated the Almagest of Ptolemy in 1160. It was identified in 1966 by John E. Murdoch.

Of course, the elements were among the first works that you wanted printed. The first Latin edition, based on the translation of Campanus of Novara, was published in 1482 in Venice. The preliminary processing of Regiomontanus was left 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 time after the invention of printing, only some important works are highlighted here: the translation of the Federicus Commandinus (1509-1575) from the Greek ( 1572), the extensively annotated edition of Christoph Clavius ​​( 1574 ) and the Oxford edition of the works of Euclid's David Gregory in 1703 ( Greek / Latin ). All of these were translations into Latin. The first translation into a modern Western language Niccolò Tartaglia published 1543. The first German translation appeared in 1555 (Johann Scheubel ) and Simon Marius in 1609 published the first German translation ( the first six books) directly from the Greek, the first English in 1570 (Henry Billingsley ), the first French 1564 ( Pierre Forcadel ), the first Spanish 1576 ( Rodrigo Zamorano ) and the first Dutch in 1606 (Jan Pieterszoon Dou ). The first translation into Chinese ( the first part ) was carried out by Xu Guangqi and Matteo Ricci ( 1607). In the popular editions from the 16th century, however, simplifications and adaptations were often made, the evidence is replaced by examples and there were also usually not published any books of the elements, but typically only the first six.

The first printed edition of the Greek text (Editio princeps ) was carried out at Simon Grynaeus 1533 in Basel.

Known comments written in antiquity, Proclus and Theon of Alexandria, who made their own arrangements in his edition of the elements. All editions known in the West until the 19th century were based on text versions descended from Theon and his school. Johan Ludvig Heiberg was only a reconstructed original version due to a manuscript ( Vaticanus Graecus 190) that was not based on the lineage of Theon, and François Peyrard discovered among the documents seized by Napoleon from the Vatican books. This Peyrard published in 1804 and 1814 to 1818 a French translation and it was used in Germany by Johann Wilhelm Camerer and Charles Frederick Hauber for their Latin- Greek edition of the first six books of 1824/25. After issuing Heiberg published Clemens Thaer a German translation ( 1933-1937 ).