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PROJECTION , in mathematics. If from a fixed point S in space lines or rays be drawn to different points A, B, C, . . . in space, and if these rays are cut by a plane in points A', B', C', .. the latter are called the projections of the given points on the plane. Instead of the plane another 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 perspective (q.v.).
In 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-ordinate 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 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 home from the ball? The whole design is notoriously full of similar incongruities, of which these are the more significant for being the most plausible. There is hardly a single work of Berlioz, except the Harold symphony and the Symphonie fantastique, in which the determination to write programme music does not frequently yield to the impulse to make singers get up and explain in words what it is all about. The climax of absurdity is in the Symphonie funebre et triomphale, written for the inauguration of the Bastille Column, and scored for an enormous military band and chorus. The first movement is a funeral 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 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 link called Oraison funebre, consisting of a sermon delivered by a solo trombone; presumably for use in later performances. His naive Gasconade genius prefers this to the use of the chorus!
Current modern criticism demands plausibility, though it cares little for intellectual soundness: and while practically the whole of Liszt's work is professedly programme music (where it is not actually vocal) and, though there is much in it which is incomplete without external explanation, Liszt is far too " modern " to betray himself into obvious confusion between different planes of musical realism . With all his unreality of style, Liszt's symphonic poems are remarkable steps towards the attainment of a kind of instrumental music which, whether its form is dictated by a programme or not, is at any rate not that of the classical symphony. The programmes of Liszt's 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 Richard Strauss are as great an advance on Liszt as they are more complex in musical, realistic and autobiographical content. Being, with the exception of the latest French orchestral developments, incomparably the most important works illustrating the present state of musical transition, they have given rise to endless discussions as to the legitimacy of programme music. Such discussions are mere windmill -tilting unless it is constantly borne in mind that no artist who has anything of his own to say will ever be prevented from saying it, in the best art-forms attain-able in his 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 general culture. (D. F. T.)
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