The Americana; a universal reference library, comprising the arts and sciences, literature, history, biography, geography, commerce, etc., of the world . H.\ of JMatlietnatics, Kenyan College, Gam-bier, Ohio. Geometry, Pure Projective. Introductory.—Projective geometr\-, as the name indicates, hasto do with the theorj of projection. Pureprojective geometry is that which is conductedby means purely geometric, without initialrecourse to algebraic methods (see Geometry,MoDERM Analytical), and which makes onlysubordinate mention of properties other thanprojective. The adjective synt
The Americana; a universal reference library, comprising the arts and sciences, literature, history, biography, geography, commerce, etc., of the world . H.\ of JMatlietnatics, Kenyan College, Gam-bier, Ohio. Geometry, Pure Projective. Introductory.—Projective geometr\-, as the name indicates, hasto do with the theorj of projection. Pureprojective geometry is that which is conductedby means purely geometric, without initialrecourse to algebraic methods (see Geometry,MoDERM Analytical), and which makes onlysubordinate mention of properties other thanprojective. The adjective synthetic is fre-quently used as practically a synonjTn forpure. The process of projection is of constantoccurrence—, in photographing (the lensmust be strictly rectilinear), in preparing alantern-slide from the photographic plate, andin throwing the image upon a screen. Thusin passing from an object to its representationupon the screen there are three successiveprojections—a fourth enters with the ^-isualimage formed upon the retina when the screenis viewed. Fig. i serv^es to illustrate theprocess of projecting a line ABCD into Fig. I. Reference to the figure will show that thelength of a line is changed by projection ABis not equal to .IS. Moreover, even theratio of two lengths is changed. AB^BC is GEOMETRY, PURE PROJECTIVE not equal to AB -hBC. The study of pro-jective geometry is the study of such propertiesof ligures as are unaltered by successive pro-jections. Lengths and ratios of lengths arenot such properties, and by right enier onlysubordinately into pure projective geometry. Historically considered, projective geometryarose by considering changes in lengths, andwas thus far from pure. A theorem attributedto Pappus () states that the double ratio ofthe lengths is unchanged by projection—thus, AB^.JLD AB AD^AB ADJiC ? VC°BC ? DC~BC • VL It was upon such basis as this that the sub-ject developed until von Staijdt () in hisfamous
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Keywords: ., bookcentury1900, bookdecade1900, booksubjectencyclo, bookyear1908
