THE ELEMENTS OF THE GEOMETRY OF PLANES AND SOLIDS. DEFINITIONS. 1. A SOLID is that which has length, breadth and thickness. 2. That which bounds a solid is a superficies. 3. A straight line is perpendicular, or at right angles to a plane, when it makes right angles with every straight line in that plane which meets it. Reciprocally, a plane is said to be perpendicular, or at right angles to a straight line, when the straight line is perpendicular to the plane. Thus the straight line AB (fig. 1), standing on the plane MN, and making right angles with every straight line, as CB, DB, EB, &c., in that plane, which meets it, is perpendicular or at right angles to the plane. Reciprocally, the plane MN is perpendicular or at right angles to the line AB. Straight lines which meet a plane and are not perpendicular to it are called oblique. 4. The inclination of a straight line to a plane, is the acute angle contained by that straight line, and another drawn from the point in which the first line meets the plane, to the point in which a perpendicular to the plane, drawn from any point of the first line above the plane, meets the same plane. Thus, the inclination of the straight line AB (fig. 2) to the plane MN is the angle contained by AB and the straight line BD drawn VOL. II. B from the point B in which AB meets the plane, to the point D in which CD perpendicular to the plane, drawn from any point C in AB, meets the plane. 5. Straight lines drawn from the same point without a plane obliquely to the plane are said to be equally distant from the perpendicular drawn from that point to the plane, when the straight lines drawn from the intersections of the oblique lines with the plane, to the intersection of the perpendicular with the plane, are equal. 6. A plane is perpendicular to a plane, when the straight lines drawn in one of the planes perpendicular to the line of intersection of the two planes are perpendicular to the other plane. Thus the plane AB (fig. 3) is perpendicular to the plane MN, when any straight lines DE, FG, HI drawn in the plane AB perpendicular to the line of intersection AC of the two planes are perpendicular to the plane MN. 7. The inclination of a plane to a plane is the acute angle contained by two straight lines drawn from any, the same point in the line of intersection of the planes, at right angles to it, one upon one plane, and the other upon the other plane. Thus the inclination of the plane BA (fig. 4) to the plane MN is the acute angle EDF contained by the two straight lines DE, DF drawn from the same point D in the line of intersection CA of the two planes, at right angles to it, DE upon the plane MN, and DF upon the plane AB. 8. Two planes are said to have the same or a like inclination to one another which two other planes have, when the said angles of inclination are equal to one another. 9. A straight line and a plane are parallel to each other when, both being indefinitely produced, they do not meet. 10. Parallel planes are such as do not meet one another, though indefinitely produced. 11. A Dihedral angle is that which is contained by two planes which cut one another. Thus the four angles which are contained by the two planes MN, AB (fig. 5), which cut one another, are Dihedral angles. A dihedral angle is expressed by four letters, one in each of the planes and two in their intersection, these last two being placed between the two others: thus the dihedral angle formed by the planes MD, AD is expressed, "the angle MCDA," and the dihedral angle formed by the two planes MD, CB is expressed, "the angle MCDB." The planes which form a dihedral angle are called its faces; and the intersection of these planes is called the edge of the dihedral angle: thus the planes MD, AD are the faces, and DC is the edge of the dihedral angle MDCA. 12. A solid angle is that which is made by the meeting of more than two plane angles, which are not in the same plane, in one point. It is called a Trihedral angle, a Tetrahedral angle, a Pentahedral angle, &c., according as it is made by three, four, five, &c. plane angles. Thus the angle which is made by the meeting in the point A (fig. 6) of the three plane angles BAC, BAD, CAD, not in the same plane, is a trihedral angle; the angle made by the meeting in the point A (fig. 7) of the four plane angles BAC, CAD, DAE, BAE, not in the same plane, is a tetrahedral angle; and both the former and the latter of these angles is a solid angle. The plane angles and also the planes which form a solid angle are called its faces; the intersections of these planes are called the edges; and the point where these planes meet is called the vertex of the solid angle. 13. Similar solid figures are such as have all their solid angles equal, each to each, and are contained by the same number of similar plane rectilineal figures. 14. A Pyramid is a solid figure contained by plane triangles that are constituted between the sides of a plane rectilineal figure and a point above it in which the vertices of all these triangles meet. 15. A Prism is a solid figure contained by plane figures, of which two that are opposite are equal, similar, and parallel to one another; and the others are parallelograms. 16. A Sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains fixed. 17. The Axis of a sphere is the fixed straight line about which the semicircle revolves. 18. The Centre of a sphere is the same with that of the semicircle. 19. The Diameter of a sphere is any straight line which passes through the centre, and is terminated both ways by the surface of the sphere. 20. A Cone is a solid figure contained by a circle and the surface described by the revolution of a straight line about the circumference, one extremity of the straight line being in a fixed point above the circle. 21. The Axis of a cone is the straight line joining the centre of the circle and the fixed point above it. 22. The Base of a cone is the circle about the circumference of which the straight line revolves. 23. A Cylinder is a solid figure described by the revolution of a right-angled parallelogram about one of its sides which remains fixed. 24. The Axis of a cylinder is the fixed straight line about which the parallelogram revolves. 25. The Bases of a cylinder are the circles described by the two revolving opposite sides of the parallelogram. 26. Similar cylinders are those which have their axes and the diameters of their bases proportionals. 27. A Cube is a solid figure contained by six equal squares. 28. A Tetrahedron is a solid figure contained by four equal and equilateral triangles. 29. An Octahedron is a solid figure contained by eight equal and equilateral triangles. 30. A Dodecahedron is a solid figure contained by twelve equal pentagons which are equilateral and equiangular. 31. An Icosahedron is a solid figure contained by twenty equal and equilateral triangles. 32. A Parallelopiped is a solid figure contained by six quadrilateral figures, whereof every opposite two are parallel. PROPOSITION I. THEOREM. One part of a straight line cannot be in a plane, and another part above it. If it be possible, let AB (fig. 8), part of the straight line ABC, be in the plane MN, and the part BC above it. Then, since the straight line AB is in the plane MN, it can be produced in that plane; let it be produced to D; and let any plane pass through the straight line AD, and be turned about it until it pass through the point C. And because the points B, C are in this plane, the straight line BC is in it (I. Def. 7)*: therefore there are two straight lines, ABC, ABD in the same plane that have a common segment AB; which is impossible (I. 11. Cor.). Therefore one part of a straight line, &c.: which was to be proved. PROP. II. THEOR. Two straight lines which cut one another are in one plane, and three straight lines which meet one another are in one plane. Let two straight lines, AB, CD (fig. 9), cut one another in E; AB, CD shall be in one plane; and three straight lines EC, CB, BE, which meet one another shall be in one plane. Let any plane pass through the straight line EB, and let the plane be turned about EB, produced if necessary, until it pass through the point C, Then because the points E, C are in this plane the straight line EC is in it (I. Def. 7): for the same reason, the straight line BC is in the same: and by the hypothesis, EB is in it therefore the three straight lines EC, CB, BE are in one plane but in the plane in which EC, EB are, in the same are CD, : *This and similar references are to the Book and Definition or Proposition in Euclid's Elements.' AB (Prop. 1)*: therefore AB, CD are in one plane which was to be proved. PROP. III. THEOR. If two planes cut one another, their common section is a straight line. Let two planes, AB, BC (fig. 10), cut one another, and let the line DB be their common section: DB shall be a straight line. If it be not, from the point D to B draw, in the plane AB, the straight line DEB (I. Post. 1), and in the plane BC, the straight line DFB. Then two straight lines DEB, DFB have the same extremities and therefore include a space betwixt them: which is impossible (I. Ax. 10): therefore BD the common section of the planes AB, BC, cannot but be a straight line: which was to be proved. PROP. IV. THEOR. If a straight line stand at right angles to each of two straight lines in the point of their intersection, it shall also be at right angles to the plane which passes through them, that is, to the plane in which they are. Let the straight line EF (fig. 11) stand at right angles to each of the straight lines AB, CD, in E, the point of their intersection: EF shall also be at right angles to the plane MN passing through AB, CD, Take the straight lines AE, EB, CE, ED, all equal to one another; join AD, CB; and through E draw, in the plane in which are AB, CD, any straight line GEH. Then from any point F, in EF, draw FA, FG, FD, FC, FH, FB. Because the two straight lines AE, ED are equal to the two BE, EC, each to each, and that they contain equal angles AED, BEC (I. 15), the base AD is equal to the base BC (I. 4), and the angle DAE to the angle EBC: and the angle AEG is equal to the angle BEH (I. 15): therefore the triangles AEG, BEH have two angles of the one equal to two angles of the other, each to each, and the sides AE, EB, adjacent to the equal angles, equal to one another : wherefore they have their other sides equal (I. 26): therefore GE is equal to EH, and AG to BH. And because AE is equal to EB, and FE common and at right angles to them, the base AF is equal to the base FB (I. 4); for the same reason, CF is equal to FD. And because AD is equal to BC, and AF to FB, the two sides FA, AD are equal to the two FB, BC, each to each; and the base DF was proved equal to the base FC; therefore the angle FAD is equal to the angle FBC (I. 8). Again, it was proved that GA is equal * This and similar references are to Propositions in these Elements. |