411 



PERSPECTIVE. 



PERSPECTIVE. 



418 



describe arc. cutting in '. Join * ', L M' ; the trknxlo ii L K' 

 luenUy (S3) reprewnU that portion o( the given plane XL n inter 

 ^plod between the co-ordinate planes and the line M m,m , brought 

 into the co-ordinate pUne by being turned round on the trace M L, and 

 by this ruUtion, bringing the original of the triangle, P A B. p a b, along 

 with it. To draw this triangle a* thus brought into the co-onlinate 

 planet, produce r B, p6, to cut the two trace* in D and e respectively, 

 make L t' equal to Lr, join DC'. In the same manner the lines a' p 

 a' b' are obtained, constituting the original triangle at brought into the 

 coordinate plane in the manner described. 



45. The poiuU A o, B 6, rp, lying in the original plane, will describe 

 arcs of circles during the rotation of that plane on its trace : the pUnes 

 of these circles must obviously be perpendicular to the original plane 

 and to the co-ordinate plane, and consequently cut the co-ordinate 

 plane in linos or traces perpendicular to M L, that of the original plane. 

 Hence if line* be drawn through A, B, and r, perpendicular to ML, they 

 will pass through a', b', and p', since the traces of these planes will be 

 the projections of all lines lying in them, and therefore of the circular 

 arcs alluded to (21). iu which the points a,', V, and p' lie. By this 

 means the points o', 6', and / may be found, or verified if previously 

 obtained on any other principle. 



46. Drawp'o' to make the proposed angle with o'b ; then the 

 plan A, and elevation of the point ', in which the proposed line meets 

 the given one in the given angle, may be determined from a' by the 

 converse proceeding to that by which a', b', and p' were obtained. 

 And lastly, P A, p a will be the projections of the line as required. 



47. The foregoing construction might have been made with the trace 

 L * instead of L M ; but the triangle, when brought into the co-ordinate 

 plane on the supposition of the rotation of the plane of that triangle on 

 L *, would not coincide with a' b'p. 



48. If a line be conceived to move always perpendicular to a co- 

 ordinate plane, and pass through an original curve, its intersection 

 with the plane will be the projection of the curve ; this projection 

 being the section of the cylindroidal surface formed by the generating 

 line If the curve be any other than a circle or an ellipse, its pro- 

 jection can only be practically described by finding the projections of 

 a sufficient number of points of the original, from some known pro- 

 perty of the curve,- or from the mode of its generation ; and the 

 required projection must be drawn by hand through the points thus 

 determined. It is obvious that the projection of any plane curve 

 which ia parallel to the co-ordinate plane must be equal and similar to 

 the original But if the original curve be a circle, or an ellipse, the 

 projecting line during its motion will generate a right or oblique cylin- 

 drical surface, the section of which by the co-ordinate plane must be 

 either a circle or an ellipse. 



49. Whatever may be the oblique position of an original circle with 

 respect to the co-ordinate plane, there must be one diameter which is 

 parallel to that plane ; now the projection of this diameter being equal 

 to the original, must be greater than those of all the other diameters 

 of the original circle, which are all necessarily oblique to the plane : 

 and since the projection of every diameter must be a diameter of the 

 projected curve, the projection of this parallel one must be the major 

 axis of the ellipse. This diameter of the original circle parallel to the 

 co-ordinate plane is that which is parallel to the trace of the plane. 

 The conjugate axis of the ellipse will be the projection of that diameter 

 * of the original circle which is perpendicular to the former, or to the 

 trace of the plane. 



60. The projection of a sphere on a co-ordinate plane must be a 

 circle of the same radius as the sphere, this circle being the projection 

 of that great circle of the solid which is parallel to the co-ordinate 



: . - 



51. In the applications of practical geometry to the arts, the object 

 is either to delineate the forms to which materials are to be reduced, 

 or to guide workmen in making and putting together machinery ; or, in 

 the construction of edifices of every description. 



62. Owing to the symmetry of the machines or edifices, the forms 

 most commonly required to be delineated are reducible to series of 

 rectangular geometrical solids, the planes of which are either parallel 

 or perpendicular to the horizon. The plans, elevations, sections, pro- 

 files, ic., furnished to the workman by the draughtsman, are the 

 projections on rectangular co-ordinate planes, assumed to be parallel 

 to the planes of the machines or edifices, made to a reduced scale ; 

 the plan being such a projection, made on a horizontal plane, and 

 the deration on a vertical plane. When the building or engine is 

 supposed to be laid open, by being cut by a plane, so as to show its 

 interior, the projection made on this supposition is termed a section, or 

 profile. 



63. It is obvious, from these assumptions, that the various plane 

 rectilinear, or mixed, figures which are produced by the intersections 

 or boundaries of the various surfaces of the original objects, are repre- 

 sented on the draughtsman's plans, Ac., by figures similar to the 

 original forms ; and that those plane surfaces of the original object 

 which are vertical to the horizon appear only as right lines on a plan, 

 bounding the figures which are the representatives of original planes 

 parallel to the horizon ; and, conversely, these last-mentioned surfaces 



In the afore, pa Is shown u the umc line u the tide of the luumcd 

 triangle, to avoid confusion ; but this, obrUmsly, need not be the oafe. 



are reutweutod by lines in the elevations, while the vertical plane 

 figures of the original are projected into similar plane figures on these 

 elevations or profiles. 



54. Henoe two, at least, and commonly three, such projections, on 

 rectangular co-ordinate planes, are requisite to convey an idea of the 

 forms of an original object ; but since these forms of the original M 

 represented of their true dimensions and proportions, such projections 

 are sufficient, and indispensable, as guides to the mechanics who are to 

 construct or put together the edifice or machine. 



65. The principles of projection enable us, as far as regards the 

 rectangular parallelopiped, the solid of most frequent occurrence, to 

 combine the two purposes for which such projections are employed ; 

 that is, to convey, by one image or figure, an accurate idea of the 

 relative position of the parallel and vertical planes of an original object, 

 reducible to this form, and at the same time to preserve one constant 

 and correct proportion between the magnitudes of the original and of 

 its representative.* 



56. It has been shown (26, 27) that the projections of definite right 

 lines, inclined in equal angles to the co-ordinate plane, will be in a 

 constant proportion to the originals; if, therefore, the three plane 



right angles forming a solid angle of a rectangular parallelopided be 

 inclined in equal dihedral angles to the 



co-ordinate plane, all lines 



parallel to the three edges of that solid angle will be projected into 

 lines bearing one constant ratio to the originals, and forming with each 

 other equal angles, which are the projections of the right ones formed 

 by the original lines. 



57. Thus, for example, if the co-ordinate plane be assumed perpen- 

 dicul ir to the diagonal of a cube, the projections of the three edges 

 meeting in either end of that diagonal will form angles of 120" with 



each other, and the three projections of the edges at one extremity 

 will, respectively, bisect the equal angles formed by those of the edges 

 at the other extremity ; while the lines joining the ends of these six 

 equal radii, which lines must obviously form a regular hexagon, will 

 be the projections of the remaining edges of the solid. Each face of 

 the cube is projected into an equilateral rhombus, as ACBF, BCDO, ACDE, 

 BFCG, 4c., the sides of which form angles respectively of 120 and of 

 60" each. If the side of the cube be unity, the equal projections of 

 those rides will be -8165, which is equal to the cosine of the angle at 

 which the originals are inclined to the co-ordinate plane. The original 

 diagonals of the three faces, AB, BD, DA, are obviously, from the 

 symmetry and position of the solid, parallel to the co-ordinate plane ; 

 their projections are therefore equal to those originals, or are each 

 equal to V2 = l'4l42 .. . If an original solid be made up of rectangular 

 parallelepipeds, having their faces mutually parallel, and the co- 

 ordinate, or plane of projection, be assumed as equally inclined to the 

 three faces forming any of the solid angles, the projections of all its 

 edges, and of all lines parallel to them, would be in the constant ratio 

 to the originals of '8164 : 1 ; the dimensions, consequently, of those 

 originals, as measured in the directions of lines which would be 

 isonietrically projected, may be set off from any scale along the 

 isometric projections of any lines parallel to the edges of the original 

 solids, and a figure or image of the original constructed which would 

 show the three principal series of planes of which that original was 

 composed. 



58. The projections of all circles equally inclined to the co-ordinate 

 plane will be similar ellipses ; the axes of these ellipses, when repre- 

 senting circles lying in planes parallel to the faces of a cube equally 

 inclined to the co-ordinate plane, will be to each other in the ratio of 

 the diagonals of the rhombus representing the inscribed or circum- 

 scribing square isometrically projected. The following simple method 

 of constructing a scale for determining the lengths of the axes of the 

 isometric projection of a circle will be of sen-ice to the practical 

 draughtsman. Construct a right-angled triangle the base aud perpen- 

 dicular of which are in the ratio of the side to the diagonal of a - 

 or as 1 : 1-4142. Set off the length of the isometric projection of the 

 circumscribing square of any original circle along the side of this 

 triangle, from the acute angle, and draw a line parallel to the other 



Thli iprcinc application of projection was termed Itomelrtc by the late 

 Profe-or Finish, wh>> pointed out Iu practical utility, and the facility of IU 

 pplloation to the delineation of engines, &c. : u a distinctive one, this tcria U 

 unexceptionable. 



