Editing History of geometry (section)

==Greek geometry==
{{see also|Greek mathematics}}

=== Thales and Pythagoras ===
[[Image:Pythagorean.svg|thumb|[[Pythagorean theorem]]: ''a''<sup>2</sup>&nbsp;+&nbsp;''b''<sup>2</sup>&nbsp;=&nbsp;''c''<sup>2</sup>]]
[[Thales]] (635–543 BC) of [[Miletus]] (now in southwestern Turkey), was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived. [[Pythagoras]] (582–496 BC) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and traveled to [[Babylon]] and [[Egypt]]. The theorem that bears his name may not have been his discovery, but he was probably one of the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of [[commensurability (mathematics)|incommensurable lengths]] and [[irrational number]]s.

=== Plato ===
[[Plato]] (427–347 BC) was a philosopher, highly esteemed by the Greeks. There is a story that he had inscribed above the entrance to his famous school, "Let none ignorant of geometry enter here." However, the story is considered to be untrue.<ref>{{cite web|last1=Cherowitzo|first1=Bill|title=What precisely was written over the door of Plato's Academy?|url=http://www.math.ucdenver.edu/~jloats/APresentations_2010/PlatosAcademy_Jim's10.ppt.pdf |archive-url=https://web.archive.org/web/20130625173419/http://www.math.ucdenver.edu/~jloats/APresentations_2010/PlatosAcademy_Jim%27s10.ppt.pdf |archive-date=2013-06-25 |url-status=live|website=www.math.ucdenver.edu/|access-date=8 April 2015}}</ref> Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but compass and straightedge – never measuring instruments such as a marked [[ruler]] or a [[protractor]], because these were a workman's tools, not worthy of a scholar. This dictum led to a deep study of possible [[compass and straightedge]] constructions, and three classic construction problems: how to use these tools to [[trisect an angle]], to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system. [[Aristotle]] (384–322 BC), Plato's greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see [[Logic]]) which was not substantially improved upon until the 19th century.

===Hellenistic geometry===

====Euclid====
[[Image:EuclidStatueOxford.jpg|thumb|Statue of Euclid in the [[Oxford University Museum of Natural History]]]]

[[Image:Woman teaching geometry.jpg|thumb|''Woman teaching geometry''. Illustration at the beginning of a medieval translation of Euclid's [[Element (mathematics)|Elements]] (c. 1310)]]
<!--[[Image:Title page of Sir Henry Billingsley's first English version of Euclid's Elements, 1570 (560x900).jpg|right|200px|thumb|The [[frontispiece]] of Sir Henry Billingsley's first English version of Euclid's ''Elements'', 1570]]-->
[[Euclid]] (c. 325–265 BC), of [[Alexandria]], probably a student at the Academy founded by Plato, wrote a treatise in 13 books (chapters), titled ''[[Euclid's Elements|The Elements of Geometry]]'', in which he presented geometry in an ideal [[axiom]]atic form, which came to be known as [[Euclidean geometry]]. The treatise is not a compendium of all that the [[Hellenistic]] mathematicians knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid's was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by [[Ptolemy I Soter|Ptolemy I]], King of Egypt.

''The Elements'' began with definitions of terms, fundamental geometric principles (called ''axioms'' or ''postulates''), and general quantitative principles (called ''common notions'') from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read.

# Any two points can be joined by a straight line.
# Any finite straight line can be extended in a straight line.
# A circle can be drawn with any center and any radius.
# All right angles are equal to each other.
# If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the [[parallel postulate]]).

Concepts, that are now understood as [[algebra]], were expressed geometrically by Euclid, a method referred to as [[Greek geometric algebra]].

====Archimedes====

[[Archimedes]] (287–212 BC), of [[Syracuse, Italy|Syracuse]], [[Sicily]], when it was a [[Greek city-state]], was one of the most famous mathematicians of the [[Hellenistic period]]. He is known for his formulation of a hydrostatic principle (known as [[Archimedes' principle]]) and for his works on geometry, including [[Measurement of the Circle]] and [[On Conoids and Spheroids]]. His work [[On Floating Bodies]] is the first known work on hydrostatics, of which Archimedes is recognized as the founder. Renaissance translations of his works, including the ancient commentaries, were enormously influential in the work of some of the best mathematicians of the 17th century, notably [[René Descartes]] and [[Pierre de Fermat]].<ref>{{cite web|publisher=Encyclopedia Britannica|url=https://www.britannica.com/biography/Archimedes|title=Archimedes}}</ref>

====After Archimedes====
[[Image:God the Geometer.jpg|thumb|left|200px|Geometry was connected to the divine for most [[History of science in the Middle Ages|medieval scholars]]. The [[Compass (drafting)|compass]] in this 13th-century manuscript is a symbol of God's act of [[Creation myth|Creation]].]]
After Archimedes, Hellenistic mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over. [[Proclus]] (410–485), author of ''Commentary on the First Book of Euclid'', was one of the last important players in Hellenistic geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek city-states produced excellent engineers, but no mathematicians of note.

The great [[Library of Alexandria]] was later burned. There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,000–70,000 scrolls in a warehouse adjacent to the port (as [[Luciano Canfora]] argues, they were likely copies produced by the Library intended for export), but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later.<ref>[[Luciano Canfora]]; ''The Vanished Library''; University of California Press, 1990. - [https://books.google.com/books?id=q6NsoT1akU4C google books]</ref>

Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in non-religious pursuits likely contributed to a reduction in the body of material available in the Library, especially in the 4th century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign.