Durer,+Albrecht+--+Renaissance+Man

His contributions to Geometry are found in his manuscripts: The Four Books on Measurement
(from Wikipedia)

//Four Books on Measurement//
===Dürer's work on [|geometry] is called the //Four Books on Measurement// (//Underweysung der Messung mit dem Zirckel und Richtscheyt// or //Instructions for Measuring with Compass and Ruler//).[|[27]] The first book focuses on linear geometry. Dürer's geometric constructions include [|helices], [|conchoids] and [|epicycloids]. He also draws on [|Apollonius], and [|Johannes Werner]'s 'Libellus super viginti duobus elementis conicis' of 1522.=== ===The second book moves onto two dimensional geometry, i.e. the construction of regular [|polygons]. Here Dürer favours the methods of [|Ptolemy] over [|Euclid].=== ===The third book applies these principles of geometry to [|architecture], [|engineering] and [|typography].=== ===In [|architecture] Dürer cites [|Vitruvius] but elaborates his own classical designs and [|columns]. In [|typography], Dürer depicts the geometric construction of the [|Latin alphabet], relying on [|Italian precedent]. However, his construction of the [|Gothic alphabet] is based upon an entirely different [|modular] system. The fourth book completes the progression of the first and second by moving to three-dimensional forms and the construction of [|polyhedra]. Here Dürer discusses the five [|Platonic solids], as well as seven [|Archimedean] semi-regular solids, as well as several of his own invention.=== ===In all these, Dürer shows the objects as [|nets]. Finally, Dürer discusses the [|Delian Problem] and moves on to the 'construzione legittima', a method of depicting a cube in two dimensions through [|linear perspective]. It was in [|Bologna] that Dürer was taught (possibly by [|Luca Pacioli] or [|Bramante]) the principles of [|linear perspective], and evidently became familiar with the 'costruzione legittima' in a written description of these principles found only, at this time, in the unpublished treatise of [|Piero della Francesca]. He was also familiar with the 'abbreviated construction' as described by Alberti and the geometrical construction of shadows, a technique of [|Leonardo da Vinci]. Although Dürer made no innovations in these areas, he is notable as the first Northern European to treat matters of visual representation in a scientific way, and with understanding of Euclidean principles. In addition to these geometrical constructions, Dürer discusses in this last book of //Underweysung der Messung// an assortment of mechanisms for drawing in perspective from models and provides woodcut illustrations of these methods that are often reproduced in discussions of perspective.===

//Four Books on Human Proportion//
===Dürer's work on [|human proportions] is called the //Four Books on Human Proportion// (//Vier Bücher von Menschlicher Proportion//) of 1528. The first book was mainly composed by 1512/13 and completed by 1523, showing five differently constructed types of both male and female figures, all parts of the body expressed in fractions of the total height. Dürer based these constructions on both [|Vitruvius] and empirical observations of, "two to three hundred living persons,"[|[18]] in his own words. The second book includes eight further types, broken down not into fractions but an [|Albertian] system, which Dürer probably learned from [|Francesco di Giorgio]'s 'De harmonica mundi totius' of 1525. In the third book, Dürer gives principles by which the proportions of the figures can be modified, including the mathematical simulation of [|convex] and [|concave mirrors]; here Dürer also deals with human [|physiognomy]. The fourth book is devoted to the theory of movement.=== ===Appended to the last book, however, is a self-contained essay on aesthetics, which Dürer worked on between 1512 and 1528, and it is here that we learn of his theories concerning 'ideal beauty'. Dürer rejected Alberti's concept of an objective beauty, proposing a relativist notion of beauty based on variety. Nonetheless, Dürer still believed that truth was hidden within nature, and that there were rules which ordered beauty, even though he found it difficult to define the criteria for such a code. In 1512/13 his three criteria were function ('Nutz'), naïve approval ('Wohlgefallen') and the happy medium ('Mittelmass'). However, unlike Alberti and Leonardo, Dürer was most troubled by understanding not just the abstract notions of beauty but also as to how an artist can create beautiful images. Between 1512 and the final draft in 1528, Dürer's belief developed from an understanding of human creativity as spontaneous or [|inspired] to a concept of 'selective inward synthesis'.[|[18]] In other words, that an artist builds on a wealth of visual experiences in order to imagine beautiful things. Dürer's belief in the abilities of a single artist over inspiration prompted him to assert that "one man may sketch something with his pen on half a sheet of paper in one day, or may cut it into a tiny piece of wood with his little iron, and it turns out to be better and more artistic than another's work at which its author labours with the utmost diligence for a whole year."[|[28]]===