INTRODUCTION. THE PLANE AND STRAIGHT LINE. DEFINITIONS. (1) A superficies is that which has only length and breadth. (2) A solid is that which has length, breadth, and thickness. (3) That which bounds a solid is a superficies. (4) A plane superficies, or a plane, is that in which any two points being taken, the straight line between them lies wholly in that superficies. (5) A straight line is said to be perpendicular, or at right angles to a plane, when it is perpendicular to every straight line meeting it in that plane. (6) The inclination of a line to a plane is the plane angle contained by that straight line, and another drawn from the point in which the first line meets the plane, through the point in which a perpendicular from any point of the first line above the plane meets the same plane. (7) The inclination of a plane to a plane is the plane angle contained by two straight lines drawn from any the same point of their common section at right angles to it, one in one plane, and the other in the other plane. (8) One plane is said to be perpendicular to another plane, when the straight lines drawn in one of the planes perpendicular to the common section of the two planes are perpendicular to the other plane. (9) Planes which do not meet one another, though pro duced, are said to be parallel. (10) The angle between two planes is called the dihedral angle. Thus, Q A B M is the dihedral angle of the two planes, QA BN, PABM: A B is called the edge, and the planes containing the angles are called faces. (11) 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. Thus, the solid angle at s is contained by the three plane angles, A S B, A SC, BSC; the planes A SB, A SC, BSC, are called faces; and s the ver tex of the solid angle. (12) A solid angle is trihedral, tetrahedral, pentahedral, &c., according as it is contained by three, four, or five, &c., faces. PROPOSITION I. One part of a straight line cannot be in a plane, and the other part above it. If possible, let a B, part of the straight line a Bc, be in the plane, and the part Bc above it. Then :: the straight line ab is in the plane, it can be produced in that plane: let it be produced to D; and let any plane pass through the straight line Ad, and revolve till it pass through the point c; then : the points B and c are in this plane; Bc is in it; :: two straight lines ABC, ABD, in the same plane, have a common segment A B, which is impossible. PROPOSITION II. Two straight lines which cut one another are in the same plane, and three straight lines which meet one another are in one plane. Let the two straight lines AB, CD, intersect at E; they shall be in the same plane; and 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 on it produced revolve round EB, till it pass through the point c. Then : E and c are in this plane, the straight line Ec is also in it; similarly bc is in it; and EB is in it by hypothesis :: EC, CB, BE, are in one plane; but in the plane in which EC, EB, are, in the same are CD, AB: :. AB and cd are in one plane. CoR. Hence a triangle EBC, or three points EBC, deter mine a plane. PROPOSITION III. If two planes cut one another, their common section is a straight line. Let two planes, A B, CD, cut one another, and let the line DB be their common section : D B shall be a straight line. For if not, from D to B, draw in the plane A B, the straight line DEB, and in the plane BC, the straight line DFB: then the two straight lines DEB, DFB, have the same extremities and :. include a space betwixt them, which is impossible: :: BD, the common section of the planes A B, BC, cannot but be a straight line. PROPOSITION IV. in the point of their intersection, it shall also be perpendi- meeting at the point B, it is also + to MN, To prove this proposition, it must be shown that A B is to any straight line BE, drawn from B, in the plane mn. Produce AB to ; make BH = AB: draw any straight line CED, cutting BC, BE, BD, in C, E, D. Join A C, A E, A D; I D, I E, I C. Then :: AB = 1 B, and ZABC = Z HBC, and Bc is common to the A 9 A B C, ABC: :: AC=1c: and similarly AD=AD. Again in AS AC D, B C D, all the sides are =; :: LACE = LACE: in AS A CE, HC E; AC=HC, CE is common, and ZACE=L HCE. AE=I E. .. in A 9 A B E, BE, AB= 1 B, Be is common, and AE = HE: LA BE= ZH BE, and they are adjacent angles. : 2 A B E is a right angle, and consequently AB is to the plane MN, PROPOSITION V. If three straight lines meet in one point, and a straight line' be perpendicular to each of them in that point, these three straight lines are in the same plane. Let the straight line AB be + to each of the straight lines BC, B D, B E, in B the point where they meet: BC, BD, BE, shall be in the same plane, If not, if possible let B D, BE, be in one plane, and let BC be above it: and let a plane pass through A B, BC, the common section of which, with the plane in which В D and BE are, is a straight line ; let this be BF. :. A B, BC, BF, are all in one plane: viz., that through A B, BC: and : A B is 1 to both BD and BE, it is t to the plane through B D and BE: and is it to B F, which is in that plane. (Prop. IV.) :: LAB F is a right Z, but _ A B C is a right 2 by hypothesis. :: Z ABF= L A B C, and they are both in the same plane, which is impossible. :: BC is not above the plane in which are BD and BE; similarly it is not below it: :. BC, BD, BE, are in the same plane. PROPOSITION VI. If two straight lines be perpendicular to a plane, they shall be parallel to each other. Let A B, C v, be 1 to the same plane; AB shall be parallel to CD. Let them meet the plane in BD: join B D: draw De 1 to B D, and in the same plane with it: make DE = AB: join B E, A E, AD. Then : AB is - to the plane in which В D, BE, are, each of the 2$, A B D, A B E, is a right 2: :: also 2 CD B, and Z CDE are right <$; and : AB=DE, and BD is common, and 2 ABD = BDE; .. AD = BE. |