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are the plane (EH' EF'), and the vertical plane FH'F Fig. 20. which is revolved into the horizontal plane. We construct L'N' and the point P', as in the preceding problem, and make upon L'P' the angle L'P'N' equal to the given angle; through the point N thus determined, we draw H'G, we then draw GF', and we have the required plane (G H', GF).
50. PROBLEM. Two lines which cut each other being given in space, to find their angle.
The two projections of the point of meeting of these two lines, will be in a straight line perpendicular to the ground line (24). Construct the given lines; find the points P', Q', (fig. 21) where they pierce the horizontal Fig. 21. plane (18); draw the straight line P'Q'; and this straight line together with the two given lines will form a triangle, of which the angle opposite to this side is the angle required.
To measure this angle, revolve the triangle about the side P'Q' into the horizontal plane. To ascertain at what distance from M' the point M will fall, suppose N to be the foot of a perpendicular drawn from M to the base P'Q' of the triangle. The line MN', which measures the height of this triangle, will be the hypothenuse of a right-angled triangle whose two sides are MM, M'N'. Revolve this triangle about the side MM' till it is parallel to the vertical plane, and the vertical projection N M' of its hypothenuse will be the height of the triangle P'MQ'. The required angle P'MQ' is now easily measured.
51. PROBLEM. The projections of a line and the traces of a plane being given, to construct their angle.
The angle which the line makes with the plane is the angle which it makes with its projection upon that plane; if, therefore, from any point in the given line, a perpendicular be drawn to the plane, it will meet this projection and a right-angled triangle will be formed ; consequently the angle which this perpendicular makes with the given line, is the complement of the angle which the given line makes with the plane. This question is therefore reduced to finding the angle of the two lines.
The angle P'MQ' (fig. 22) is the complement of the Fig. 22. required angle, whose construction will be readily under
stood from the last article, observing that (OK', OK") represents the proposed plane, and that the other parts of the construction are the same as in the preceding solution.
52. PROBLEM. Two straight lines which do not cut each other being given in space, to find the distance between their nearest points.
Suppose first that one of the given straight lines is perFig. 23. pendicular to the horizontal plane (fig. 23), its projection
in this plane will be the point M' (16), and its vertical projection will be MM" perpendicular to the ground line. Draw M'P' perpendicular to the horizontal projection of the second given line, and this will be the distance sought.
53. The shortest distance between two straight lines, Fig. 21. M'N' and EF (fig. 24) may also be found, by drawing
through one of them a plane H G' parallel to the second, and then drawing from any point in the second a perpendicular EF' to this plane. This perpendicular is the distance sought, and determines the plane FEE which cuts the straight line M'N' in the point P', where this straight line approaches the nearest to EF.
Let the line (EP', E'P') (fig. 25) be the first of these lines, and (OM, O'M) the second. E' is the point wliere the first given line pierces the horizontal plane; and (E'L', EL") is a line drawn through E' parallel to the second given line, to determine a plane parallel to this last. The plane passing through the line drawn as above, and through the first of the given straight lines, is constructed by a process analogous to that of article 44; this plane is G GG"; and as it is parallel to the second of the given straight lines, it is only requisite to draw, through any point of this last, a perpendicular to this plane ; this is done through the point 0 0'.
We next seek the point of meeting of this perpendicular with the plane G'GG" (33); this point is (N', N').
In order to find the nearest points of the proposed lines, we draw, through the point N', (parallel to M O the horizontal projection of the second straight line) the line N'P', which is evidently the horizontal projection of the intersection of the plane G'GG'' with a plane drawn perpendicular to this and through the second given line, since it belongs to a straight line parallel to this, and which passes through the foot of the perpendicular Fig. 25. drawn from this straight line to the plane in question ; in other words, it is the projection of E' F' in figure 24.
The point P' in which the line N'P' meets the horizontal projection E' P' of the first of the given lines, is the projection of the point P' of figure 24, and consequently the projection of the point where the first straight line approaches the nearest possible to the second. The perpendicular to the plane GÄGG", drawn through this point, is the shortest distance sought; its projections are P'K', P"K", parallel to N'O', N" O, respectively ; its length is the hypothenuse of a right-angled triangle, of which one side is P'K', and the other, the difference of the distances of the points P" and K" above the ground line. This will be found by article 21.
54. Problem. Having two of the plane angles in any triedral angle, and the inclination of these planes, to construct the developement of this triedral angle.
Suppose one of the given faces Af'f to revolve into the plane of the other AfE (fig. 26); if through any point f, Fig. 26. taken at pleasure in the edge Af, a perpendicular ff' be drawn, it will describe in this developement, a plane perpendicular to this edge. In this plane, the angle of inclination of the given planes will be found; if, therefore, we make upon f F the angle F"f F equal to the given angle, and take f F" equal to ff', we shall thus have the situation of the point f' in relation to the line f F, when the planes have their given inclination. But it is evident that the three points fF" and F, determine the base of a pyramid formed by the proposed triedral angle and the plane which has been drawn perpendicular to the edge Af; and the required face of this triedral angle will meet the first face in AF, and the second in Af, aud will apply itself to the constructed triangle, in the line FF". The third face will therefore be determined by the triangle whose three sides are AF, Af' and FF".
Remark. This is one case of the general problem Any three of the six things in 2 triedral angle (namely, the three plane angles and the three diedral angles) being given, to find the others. As every spherical triangle answers to a triedral angle at the centre of the sphere, (the sides of the spherical triangle measuring the plane angles in the triedral angle, and the angles of
Fig. 26. the triangle being the diedral angles in the triedral an
gle,) it is manifest that the solution of this general problem is the solution of all problems which can occur in spherical trigonometry.
As all polyedral bodies are determined by planes and their intersections, the preceding problems will enable us to construct all kinds of polyedrons.
SECTION III.Of the Generation of Geometrical
Magnitudes. 55. If we conceive a point to move constantly in the same direction, it describes a straight line. The straight line is the path described by the moving point; that is, it consists of the successive positions occupied by this point. And these positions, gr points, we call the elements of the line; and we say that the line is generated by the motion of the point.
It will be readily seen that the number of these elementary positions in a finite line, is illimitable; we therefore say, that the line is composed of an infinite number of points.
If a straight line be conceived to move in such a manner that any two points in it shall describe equal and parallel straight lines, it will be always in the same plane, and will generate a parallelogram. This surface will be composed of the successive positions of this straight line; and these positions or lines constitute the elements of the surface. And all plane surfaces may be considered as composed of right-line or rectilinear ele
A circle may be considered as composed of an infifinite number of parallel straight lines. It must then be considered to have been generated by a straight line (its diameter) moving in a plane and changing its magnitude with its distance each way from its first position, according to a certain law.
Or, as a circle is generated by the motion, in a plane, of a straight line equal to radius, about one extremity (El. 266); each elementary point in this moveable radius may be considered as generating a circular curve. These circular curves compose the entire surface of the circle, and constitute its curvilincar or circular elements.
56. The motion of a surface, out of the plane of its original position, generates a body.
If a square move in such a manner that any three points in it, not in the same straight line, shall describe parallel straight lines, each equal to a side of the square and perpendicular to the surface, the product of this motion will be a cube ; and the successive positions of this square are the elements of the cube.
İf the several points in the square should move in straight lines passing through the same fixed point, the dimensions of the square will be in a constant ratio to its distance from that point; and must therefore all be zero (0) at the instant of passing that point. The body thus generated by the moving square will be a pyramid whose summit is the fixed point, whose base is the square, and whose elements are the successive positions and magnitudes of this square.
A triangular or any other polygonal pyramid may be conceived to be generated in the same manner. The analogous method of generating the cone and cylinder, was noticed in the Elements (El. 266, 275).
A sphere may also be considered as generated by the direct motion of its great circle always parallel to itself; its centre describing a straight line perpendicular to its plane; and its area diminishing with its distance (each way) from its original position, according to a certain law.
In these examples, every successive position of the generating plane is a section of the body thus generated. And as these successive positions constitute the whole intermediate space, and are all parallel, the body may be considered as composed of an infinite number of parallel sections.
57. If a point in motion continually change its direction, its path is a curve line. If this line lie wholly in the same plane, it is called a curve of single curvature.
If the points of the curve are not in the same plane, it is called a curve of double curvature.
58. Mathematical curves are considered to be given, when the law, according to which the generating point changes its direction, is given.
A point in motion will generate a circular curve, when