Editing History of geometry (section)

==Renaissance==
[[Image:Durer astronomer.jpg|thumb|225px|An engraving by [[Albrecht Dürer]] featuring [[Mashallah ibn Athari|Mashallah]], from the title page of the ''De scientia motus orbis'' (Latin version with engraving, 1504). As in many medieval illustrations, the [[Compass (drafting)|compass]] here is an icon of religion as well as science, in reference to God as the architect of creation.]]

The [[transmission of the Greek Classics]] to medieval Europe via the [[Islamic contributions to Medieval Europe|Arabic literature]] of the 9th to 10th century "[[Islamic Golden Age]]" began in the 10th century and culminated in the [[Latin translations of the 12th century]].
A copy of [[Ptolemy]]'s ''[[Almagest]]'' was brought back to Sicily by [[Henry Aristippus]] (d. 1162), as a gift from the Emperor to [[William I of Sicily|King William I]] (r. 1154&ndash;1166). An anonymous student at Salerno travelled to Sicily and translated the ''Almagest'' as well as several works by Euclid from Greek to Latin.<ref>[[Marie-Thérèse d'Alverny|d'Alverny, Marie-Thérèse]]. "Translations and Translators", in Robert L. Benson and Giles Constable, eds., ''Renaissance and Renewal in the Twelfth Century'',  421–462. Cambridge: Harvard Univ. Pr., 1982, pp. 433–4.</ref> Although the Sicilians generally translated directly from the Greek, when Greek texts were not available, they would translate from Arabic. [[Eugenius of Palermo]] (d. 1202) translated Ptolemy's ''[[Ptolemy#Optics|Optics]]'' into Latin, drawing on his knowledge of all three languages in the task.<ref>M.-T. d'Alverny, "Translations and Translators," p.&nbsp;435</ref> The rigorous deductive methods of geometry found in Euclid's ''Elements of Geometry'' were relearned, and further development of geometry in the styles of both Euclid ([[Euclidean geometry]]) and Khayyam ([[algebraic geometry]]) continued, resulting in an abundance of new theorems and concepts, many of them very profound and elegant.

Advances in the treatment of [[Perspective (graphical)|perspective]] were made in [[Renaissance art]] of the 14th to 15th century which went beyond what had been achieved in antiquity. 
In [[Renaissance architecture]] of the ''[[Quattrocento]]'', concepts of architectural order were explored and rules were formulated. A prime example of is the [[Basilica di San Lorenzo di Firenze|Basilica di San Lorenzo]] in [[Florence]] by [[Filippo Brunelleschi]] (1377–1446).<ref>Howard Saalman. ''Filippo Brunelleschi: The Buildings''. (London: Zwemmer, 1993).</ref>

In c. 1413 [[Filippo Brunelleschi]] demonstrated the geometrical method of perspective, used today by artists, by painting the outlines of various [[Florence|Florentine]] buildings onto a mirror. 
Soon after, nearly every artist in Florence and in Italy used geometrical perspective in their paintings,<ref>"...and these works (of perspective by Brunelleschi) were the means of arousing the minds of the other craftsmen, who afterwards devoted themselves to this with great zeal."<br>Vasari's ''Lives of the Artists'' Chapter on Brunelleschi</ref> notably [[Masolino da Panicale]] and [[Donatello]]. [[Melozzo da Forlì]] first used the technique of upward foreshortening (in Rome, [[Loreto (AN)|Loreto]], [[Forlì]] and others), and was celebrated for that. Not only was perspective a way of showing depth, it was also a new method of [[Composition (visual arts)|composing]] a painting. Paintings began to show a single, unified scene, rather than a combination of several.

As shown by the quick proliferation of accurate perspective paintings in Florence, Brunelleschi likely understood (with help from his friend the mathematician [[Paolo dal Pozzo Toscanelli|Toscanelli]]),<ref>"Messer Paolo dal Pozzo Toscanelli, having returned from his studies, invited Filippo with other friends to supper in a garden, and the discourse falling on mathematical subjects, Filippo formed a friendship with him and learned geometry from him."<br>Vasarai's ''Lives of the Artists'', Chapter on Brunelleschi</ref> but did not publish, the mathematics behind perspective. Decades later, his friend [[Leon Battista Alberti]] wrote ''[[De pictura]]'' (1435/1436), a treatise on proper methods of showing distance in painting based on Euclidean geometry. Alberti was also trained in the science of optics through the school of Padua and under the influence of [[Biagio Pelacani da Parma]] who studied Alhazen's ''Optics''.

[[Piero della Francesca]] elaborated on Della Pittura in his ''[[De Prospectiva Pingendi]]'' in the 1470s. Alberti had limited himself to figures on the ground plane and giving an overall basis for perspective. Della Francesca fleshed it out, explicitly covering solids in any area of the picture plane. Della Francesca also started the now common practice of using illustrated figures to explain the mathematical concepts, making his treatise easier to understand than Alberti's. Della Francesca was also the first to accurately draw the [[Platonic solids]] as they would appear in perspective.

Perspective remained, for a while, the domain of Florence. [[Jan van Eyck]], among others, was unable to create a consistent structure for the converging lines in paintings, as in London's [[The Arnolfini Portrait]], because he was unaware of the theoretical breakthrough just then occurring in Italy. However he achieved very subtle effects by manipulations of scale in his interiors. Gradually, and partly through the movement of academies of the arts, the Italian techniques became part of the training of artists across Europe, and later other parts of the world.
The culmination of these Renaissance traditions finds its ultimate synthesis in the research of the architect, geometer, and optician [[Girard Desargues]] on perspective, optics and projective geometry.

The ''[[Vitruvian Man]]'' by [[Leonardo da Vinci]] (c. 1490)<ref>The Secret Language of the Renaissance - Richard Stemp</ref> depicts a man in two superimposed positions with his arms and legs apart and inscribed in a circle and square. The drawing is based on the correlations of ideal [[Body proportions|human proportions]] with geometry described by the ancient Roman architect [[Vitruvius]] in Book III of his treatise ''[[De Architectura]]''.