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remainder-expression: *
Illustration- this is an example of expression for the sum of a convergent series:
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The reciprocals of Fibonacci numbers produce a convergent series. The ratio of successive terms in this sum tends to the reciprocal of the golden ratio. Since this is less than 1, the ratio test shows that the sum converges.
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Fouquet's Vaux-le-Vicomete:
Built 1658-61, an influential work of architecture; the architect Louis Le Vau, le Nôtre & painter-decorator Charles Le Brun collaborated for the first time. It marked the beginning of the "Louis XIV style" combining architecture, interior design & landscape. The garden's pronounced visual axis is an example of this style.
In 1641 Fouquet purchased the small château adjacent to royal residences of Vincennes & Fontainebleau. In 1656, he commissioned Le Vau, Le Brun & Le Nôtre to renovate his estate to match his ambitions. He purchased & demolished 3 villages to allow for the elaborate plans. The château rises on an elevated platform in the middle of the woods, marking the border between unequal spaces, each treated in a different way. The garden was the dominant structure, stretching 1.5 miles with a balanced composition of water basins & canals contained in stone curbs, fountains, gravel walks & patterned parterres. The site was naturally well-watered, with 2 small rivers that meet in the park; the canalized bed of forms the Grand Canal leading to a square basin. Le Nôtre created a magnificent scene to be viewed from the house, the natural terrain used to advantage, placing the canal at the lowest part of the complex, hiding it from the main perspective point of view. Past the canal, the garden ascends to a large open lawn. Shrubberies provided a picture frame to the garden which also was a stage for royal fêtes. Le Nôtre employed anamorphosis abscondita (translated as 'hidden distortion', a type of optical illusion) to establish decelerated perspective. The most apparent element in this is the reflecting pools, narrower at the closest point to the viewer (standing at the rear of the château) than at their farthest point; thus appearing closer to the viewer. From a certain designed viewing point, the distortion designed into the landscape elements produces a particular forced perspective; the eye perceives the elements closer than they actually are. This designed viewing point is at the top of the stairs at the rear of the château; here one experiences the garden with a magnificent perspectival view. The anamorphosis abscondita creates visual effects (not encountered in nature), making the gardens unusual (the viewer experiences a tension between the natural perspective cues in his peripheral vision & the forced perspective of the formal garden). From here the entire garden is revealed in a single glance. Initially, the view consists of symmetrical rows of shrubbery, avenues, fountains, statues, flowers & other pieces developed to imitate nature which exemplify the Baroque desire to mould nature to fit its wishes, using nature to imitate nature. The centrepiece is a large reflecting pool flanked by grottos holding statues in their many niches. The grand sloping lawn is not visible until one begins to explore the garden, when the viewer is made aware of the optical elements involved & discovers the garden is much larger than it looks. Next, a circular pool (previously seen as ovular due to foreshortening) is passed & a canal that bisects the site is revealed, as well as a lower level path. As the viewer continues on, the second pool shows itself to be square, the grottos & their niche statues become clearer. However, when one walks towards the grottos, the relationship between the pool and the grottos appears awry as the grottos are actually on a much lower level than the rest of the garden & separated by a wide canal that is over half a mile long. This optical effect is a result of the use of the 10th theorem of Euclid's Optics: the most distant parts of planes below the eye appear to be the most elevated. Interested parties could cross the canal in a boat, but walking around the canal provides a view of the woods that mark what is no longer the garden & shows the distortion of the grottos previously seen as sculptural. Once canal & grottos are passed, the large sloping lawn is reached & the garden is viewed from the initial viewpoint's vanishing point, completing the circuit as intended by Le Nôtre. From here the distortions create the illusion that the gardens are much longer than they actually are.
And see illustration
view from gardens to The Château
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Renaissance park (Medici):
The oldest existing Italian Renaissance garden is at the Villa Medici in Fiesole, north of Florence, created 1455-61 by Giovanni de' Medici (1421–1463). The villa was located on a rocky hillside with a view over Florence. It followed Alberti's precepts that a villa should have a view that overlooks the city & that the foreground have delicate gardens. It has 2 large terraces, 1 at the ground floor level and 1 other at the level of the first floor. From the reception rooms on the first floor, guests could go out to the loggia and from there to the garden so the loggia was a transition space connecting the interior with the exterior. It did not have a grand staircase to link the two levels. The garden was inherited by his nephew, Lorenzo de' Medici, who made it a meeting place for poets, artists, writers and philosophers.
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Macpherson's" Ossian" (autumnal passing, approaching end): *
In 1761 he announced the discovery of an epic on the subject of Fingal (related to the Irish mythological character Finn McCool) written by Ossian. That same year he published “Fingal, an Ancient Epic Poem in Six Books”, together with Several Other Poems composed by Ossian, Son of Fingal, all translated from the Gaelic Language. It was written in the musical measured prose of which he had made used in an earlier published work. Another Ossian based publication Temora followed in 1763, & then a collected edition, The Works of Ossian, in 1765. The authenticity of these translations of a 3rd-century bard was immediately challenged by Irish historians; technical errors in chronology & in the forming of Gaelic names were noted as well as the sheer implausibility of many of Macpherson's claims. Macpherson never produced the originals that he claimed existed. More forceful denunciations were later made by Samuel Johnson, who denounced Mcphereson & asserted in A Journey to the Western Islands of Scotland (1775) that Macpherson had found fragments of poems and stories, and then woven them into a romance of his own composition. Malcolm Laing, in an appendix to his History of Scotland (1800), concluded that the so-called Ossianic poems were modern in origin, and that Macpherson's authorities were non-existent.
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Park of Versailles:
Grand perspective looking to the Latona Fountain with the tapis vert and the Grand Canal in the background
This topographic map (below) clearly defines the main east-west and north-south axis that anchors the gardens’ layout. The grand perspective or view is shown with the blue symbol.
