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PROJECTION

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Originally appearing in Volume V22, Page 427 of the 1911 Encyclopedia Britannica.
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PROJECTION , in See also:

mathematics. If from a fixed point S in space lines or rays be See also:drawn to different points A, B, C, . . . in space, and if these rays are cut by a See also:plane in points A', B', C', .. the latter are called the projections of the given points on the plane. Instead of the plane another See also:surface may be taken, and then the points are projected to that surface instead of to a plane. In this manner any figure, plane or in space of three dimensions, may be projected to any surface from any point which is called the centre of projection. If the figure projected is in three dimensions then this projection is the same as that used in what is generally known as See also:perspective (q.v.). In See also:modern mathematics the word projection is often taken with a slightly different meaning, supposing that plane figures are projected into plane figures, but three-dimensional ones into three-dimensional figures. Projection in this sense, when treated by co-See also:ordinate See also:geometry, leads in its algebraical aspect to the theory of linear substitution and hence to the theory of invariants and co-variants (see ALGEBRAIC FORMS). In this See also:article projection will be treated from a purely geometrical point of view. References like (G. § 87) relate to the article GEOMETRY, § Projective, in vol. xi. degrees of conviviality returning See also:home from the See also:ball?

The whole See also:

design is notoriously full of similar incongruities, of which these are the more significant for being the most plausible. There is hardly a single See also:work of See also:Berlioz, except the Harold See also:symphony and the Symphonie fantastique, in which the determination to write See also:programme See also:music does not frequently yield to the impulse to make singers get up and explain in words what it is all about. The See also:climax of absurdity is in the Symphonie funebre et triomphale, written for the inauguration of the See also:Bastille See also:Column, and scored for an enormous military See also:band and See also:chorus. The first See also:movement is a funeral See also:march, and is not only one of Berlioz's finest pieces, but probably the greatest work ever written for a military band. The A potheose chorus is in the See also:form of a triumphal march. Because the occasion was one on which there would be plenty of real speeches, Berlioz must needs write a connecting See also:link called Oraison funebre, consisting of a See also:sermon delivered by a See also:solo See also:trombone; presumably for use in later performances. His naive Gasconade See also:genius prefers this to the use of the chorus! Current modern See also:criticism demands plausibility, though it cares little for intellectual soundness: and while practically the whole of See also:Liszt's work is professedly programme music (where it is not actually vocal) and, though there is much in it which is incomplete without See also:external explanation, Liszt is far too " modern " to betray himself into obvious confusion between different planes of musical See also:realism. With all his unreality of See also:style, Liszt's symphonic poems are remarkable steps towards the attainment of a See also:kind of instrumental music which, whether its form is dictated by a programme or not, is at any See also:rate not that of the classical symphony. The programmes of Liszt's See also:works have not always, perhaps not often, produced a living musical form; a form, that is, in which the rhythms and proportions are neither stiff nor nebulous. Both in breadth of design and in organization and flow, the works of See also:Richard See also:Strauss are as See also:great an advance on Liszt as they are more complex in musical, realistic and autobiographical content. Being, with the exception of the latest See also:French orchestral developments, incomparably the most important works illustrating the See also:present See also:state of musical transition, they have given rise to endless discussions as to the See also:legitimacy of programme music.

Such discussions are See also:

mere See also:windmill-tilting unless it is constantly See also:borne in mind that no artist who has anything of his own to say will ever be prevented from saying it, in the best See also:art-forms attain-able in his See also:day, by any scruples as to whether the antecedents of his art-forms are legitimate or not. There is only one thing that is artistically legitimate, and that is a perfect work of art. And the only thing demonstrably prejudicial to such legitimacy in .a piece of programme music is that even the most cultured of musicians generally understand music better than they under-stand anything else, while the greatest musicians know more of their art than is dreamt of in See also:general culture. (D. F.

End of Article: PROJECTION

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