19. In fig. 3 are represented circles as projected in ellipses on the isometric planes with their major and minor axes, and isometric diameters. If such circles represent wheels in machinery, or circles having axles in an instrument, these axles being perpendicular to the isometric planes, must be in the direction of the minor axes of the ellipses. 20. If the circle to be represented be graduated, that is, have its circumference divided into any number of equal parts, these graduations may be correctly represented on the ellipse. Let AG (fig. 4) be the axis major of the ellipse AeG representing the circle upon the isometric plane; then the circle AEG described upon AG as a diameter, will be that circle in its true dimensions (17). Let B, C, D be the points of graduation in the circle: then since the diameter AG is parallel to the plane of projection, the projections of B, C, D will be in planes perpendicular to AG, and passing through B, C, D; and consequently will be b, c, d, the intersections of the perpendiculars to AG drawn from B, C, D, with the ellipse AeG representing the circle on the plane of projection. 21. By means of a series of these ellipses, we obtain scales on which distances in other directions than the isometric may be measured. For this purpose it is only necessary to take the intervals between the ellipses, in the isometric directions from the centre, equal to the units on the isometric scale, as in fig. 5; the intersections of these ellipses with any line drawn through the centre will be at intervals which are the units on that line, and will therefore form the scale for that line and for all lines which have the same position with reference to the isometric lines. Thus distances measured on the longer diagonals AF, FD, DA (fig. 1) of these squares, or on lines parallel to them, are to be measured by the divisions on the major axis; and those on the shorter diagonals OG, OE, OB, or their parallels, by divisions on the minor axis: and so in other directions. axes, 22. For the purpose of drawing lines parallel to the isometric and at required distances from a regulating point, a long flat rectangular ruler, having its edges on each side graduated from their middle points and to different scales, with an equilateral triangle, having an index mark in the middle of each side, may be most conveniently employed. 23. In applying the principles of isometric perspective to the representation of objects, it is necessary to bear in mind the assumphaving a given isometric diameter, the distance between the pins must be taken equal to the given diameter, and the length of the string must be to the distance between the pins as 1+6 is to 1, or as 2.225 to 1. In practice it may be best to double the string, and tie it so that the length to the knot shall be to half the distance between the pins, as 2.225 to 1. tion which may be made regarding the scale to which the drawing is adapted, in conformity with what has already been stated (7). When the drawing is to be considered merely as representing the object to a scale, for instance to one of of the real dimensions, then dimensions represented in the directions of the isometric axes are to be taken on a scale of, and applied at once to those axes. In this case the isometric diameters of any ellipses will be the diameters of the circles they represent taken on the scale of When, however, the drawing is to be considered as the isometric projection of a model of the object on a reduced scale of T, then dimensions in the directions of the isometric axes are to be taken 1 1 on a scale of 6. Such a scale may be readily constructed 100 3 by drawing a horizontal line, and another inclined to it at an angle of 35° 16'; constructing the given scale on the inclined line, and from the points of division letting fall perpendiculars on the horizontal line the horizontal scale will be the one required. The manner of applying the principles of isometric perspective to the representation of objects will be sufficiently illustrated by fig. 6 (Pl. XII.), which represents a Powder Magazine. GEOMETRY OF PLANES AND SOLIDS. (CONTINUED.) GEOMETRY OF SOLIDS. PROP. XLIX. THEOR. Two solids which are contained by the same number of equal and similar plane figures, similarly situated, and which have none of their solid angles contained by more than three plane angles, are equal and similar to one another. Let the solids AI, NW (fig. 54), which have none of their solid angles contained by more than three plane angles, be contained by the same number of equal and similar plane figures similarly situated, viz. the plane figure ABCDE equal and similar to NOPQR; also ABGFM to NOTSZ; BH to OV; CI to PW; DK to QX; EKLMA to RXYZN; and LMF to YZS: the solid AI shall be equal and similar to the solid NW. Since the plane figures which contain the two solids are equal, similar and similarly situated, the plane angles which contain the solid angles of the one are respectively equal to the plane angles which contain the corresponding solid angles in the other: the angles ABC, ABG, GBC severally equal to the angles NOP, NOT, TOP; and the same of the angles which contain the other solid angles. And since the plane angles which contain the solid angle B are equal to the plane angles which contain the solid angle O, the dihedral angles contained by the faces about the angle B, will be equal to the dihedral angles contained by the equal faces about the angle O (Prop.47). If, therefore, the solid AI be applied to the solid NW, so that the plane figure ABCDE may coincide with NOPQR to which it is equal and similar, the point B coinciding with O, and AB, BC coinciding with NO, OP to which they are equal, the plane figures ABGFM and GBCH will coincide with NOTSZ and TOPV to which they are respectively equal and similar, the dihedral angles formed by the planes ABGFM and GBCH with the plane ABCDE, and with each other, being respectively equal to the dihedral angles formed by the planes NOTSZ and TOPV, with the plane NOPQR, and with each other. In the same manner it may be shown that VOL. II. I the figure HCDI will coincide with VPQW; IDEK with WQRX; KEAML with XRNZY; FGHIKL with STVWXY; and LMF with YZS and the solid AI will coincide with the solid NW, and be equal to it: which was to be proved. PROP. L. THEOR. Two prisms which have a solid angle in the one, contained by three plane figures, equal, similar and similarly situated to the three plane figures about a solid angle of the other, are equal and similar. Let the two prisms AI, LT (fig. 55), have the solid angle at B, in the one, contained by the three plane figures ABCDE, ABGF, GBCH equal, similar and similarly situated to the three plane figures LMNOP, LMRQ, RMNS about the solid angle at M, in the other; the prism AI shall be equal and similar to the prism LT. Because the plane fignres about the solid angle at B are equal, similar and similarly situated to the figures about the solid angle at M, the three plane angles of the one solid angle are equal to the three plane angles of the other, each to each, and therefore the dihedral angles contained by the faces of the one are equal to the dihedral angles contained by the faces equal to them of the other (Prop. 47), the angle EABG equal to the angle PLMR, the angle ABCH equal to the angle LMNS and the angle ABGH equal to the angle LMRS. And because the plane angles BAE, BAF, of the solid angle at A, are equal to the plane angles MLP, MLQ, of the solid angles at L, each to each, and the dihedral angles EABG, PLMR, formed by the planes, are also equal, the third plane angles FAE, QLP of the solid angles at A and L are equal (Prop. 48). The parallelograms AK, LV have therefore the two sides FA, AE equal to the two sides QL, LP, each to each, and the angle FAE equal to the angle QLP; consequently the parallelogram AK is equal and similar to the parallelogram LV. In the same manner it may be shown that the parallelogram EI is equal and similar to the parallelogram PT, and the parallelogram DH equal and similar to the parallelogram OS. And because the figure FGHIK is equal and similar to ABCDE (Def. 15), and QRSTV equal and similar to LMNOP, and that ABCDE and LMNOP are equal and similar, FGHIK is equal and similar to QRSTV (VI. 21). Since, therefore, the prisms AI and LT are contained by the same number of equal and similar plane figures similarly situated, they are equal and similar to one another (Prop. 49): which was to be proved. Cor. It follows from this that if two prisms AI, LT upon equal and similar bases ABCDE, LMNOP, have a bounding parallelogram AG standing upon the side AB of the base of the one, equal and similar, and similarly situated to the bounding parallelogram LR standing upon the equal and homologous side LM of the base of the other, and have the dihedral angles EABG, PLMR, contained by these parallelograms and the bases also equal, they are equal and similar to one another; for in this case the angle FAE will be equal to the angle QLP (Prop. 48), and therefore the parallelogram AK equal, similar, and similarly situated to the parallelogram LV. PROP. LI. THEOR. If a solid be contained by six planes, two and two of which are parallel, the opposite planes are similar and equal parallelograms. Let the solid ABFE (fig. 56) be contained by the parallel planes AC, GF; BG, CE; FB, AE: its opposite planes shall be similar and equal parallelograms. Because the two parallel planes BG, CE, are cut by the plane AC, their common sections AB, CD, are parallel (Prop. 27): again, because the two parallel planes BF, AE are cut by the plane AC, their common sections AD, BC are parallel (Prop. 27): and AB is parallel to CD; therefore AC is a parallelogram. In like manner, it may be proved that each of the figures CE, FG, GB, BF, AE, is a parallelogram. Join AH, DF; and because AB is parallel to DC, and BH to CF, the two straight lines AB, BH, which meet one another, are parallel to DC and CF, which meet one another: wherefore they contain equal angles (Prop. 20); therefore the angle ABH is equal to the angle DCF. And because AB, BH, are equal to DC, CF, each to each, and the angle ABH equal to the angle DCF; therefore the base AH is equal to the base DF (I. 4), and the triangle ABH to the triangle DCF : but the parallelogram BG is double of the triangle ABH (I. 34), and the parallelogram CE double of the triangle DCF; therefore the parallelogram BG is equal and similar to the parallelogram CE. In the same manner it may be proved, that the parallelogram AC is equal and similar to the parallelogram GF, and the parallelogram AE to BF. Therefore, if a solid, &c.: which was to be proved. PROP. LII. THEOR. The opposite dihedral angles of a parallelepiped are equal; and its opposite trihedral angles are symmetrical. Let ABFE (fig. 56) be a parallelepiped; its opposite dihedral angles are equal, viz. the dihedral angles GABC and CFEG; GADC and CFHG; BAGE and EFCB; ABHF and FEDA; ABCF and FEGA; AGHF and FCDA; and its opposite trihedral angles are symmetrical, viz. the solid angles at A and F; at C and G; at D and H; and at B and E. |