ELEMENTS OF GEOMETRY. BOOK VII. DEFINITION S. A I. straight line is perpendicular, or at right angles to a Book VII. plane, when it makes right angles with every straight line meeting it in that plane. II. A plane is perpendicular to a plane, when the ftraight lines drawn in one of the planes perpendicularly to the common section of the two planes, are perpendicular to the other plane. III. The inclination of a straight line to a plane is the acute angle contained by that ftraight line, and another drawn from the point in which the first line meets the plane, to the point 2 Book VII. point in which a perpendicular to the plane drawn from any point of the first line above the plane, meets the fame plane. IV. The inclination of a plane to a plane is the acute angle contained by two straight lines drawn from any the fame point of their common fection at right angles to it, one upon one plane, and the other upon the other plane. Two planes are said to have the fame, or a like inclination to one another, which two other planes have, when the faid angles of inclination are equal to one another. VI. Parallel planes are fuch as do not meet one another though produced. VII. A Solid is that which has length, breadth, and thickness. See N. A folid angle is that which is made by the meeting of more than two plane angles, which are not in the fame plane, in one point. IX. See N. Similar folid figures are fuch as are contained by the fame number of fimilar planes fimilarly fituated, and having like inclinations to one another.! X. A pyramid is a solid figure contained by planes that are conftituted betwixt one plane and a point above it in which they meet. XI. XI. A prifm is a folid figure contained by plane figures, of which two that are oppofite are equal, fimilar, and parallel to one another; and the others parallelograms. XII. A parallelopiped is a folid figure contained by fix quadrilateral figures, whereof every opposite two are parallel. XIII. A cube is a folid figure contained by fix equal fquares. XIV. A sphere is a folid figure defcribed by the revolution of a semicircle about a diameter, which remains unmoved. XV. The axis of a sphere is the fixed straight line about which the femicircle revolves. Book VII. XVI. The centre of a sphere is the same with that of the femicircle. XVII. The diameter of a sphere is any ftraight line which paffes through the centre, and is terminated both ways by the fuperficies of the sphere. XVIII. A cone is a folid figure described by the revolution of a right angled triangle about one of the fides containing the right angle, which fide remains fixed. XIX. The axis of a cone is the fixed straight line about which the triangle revolves. XX. Book VII. XX. The base of a cone is the circle defcribed by that fide containing the right angle, which revolves. XX. A cylinder is a folid figure described by the revolution of a right angled parallelogram about one of its fides which remains fixed. XXI. The axis of a cylinder is the fixed straight line about which the parallelogram revolves. XXII. The bafes of a cylinder are the circles defcribed by the twe revolving oppofite fides of the parallelogram. XXIII. Similar cones and cylinders are those which have their axes and the diameters of their bafes proportionals. Book VII. Ο PROP. I. THE OR. NE part of a ftraight line cannot be in a plane see N. and another part above it. If it be poffible, let AB, part of the straight line ABC, be in the plane, and the part BC above it: and fince the straight line AB is in the plane, it can be produced in that plane: let it be produced to D: Then ABC and ABD are two ftraight lines, and they have the common fegment AB, which is impoffible b. There B fore ABC is not a straight line. Wherefore one part, &c. Q. E. D. b cor, def. 4. I. A PROP. II. THEOR. NY three ftraight lines which meet one ano- Let the three straight lines AB, CD, CB, meet one another in the points B, C and E; AB, CD, CB are in one plane. Let any plane pafs through the ftraight line EB, and let the plane A be turned about EB, produced, if neceffary, until it pafs through the point C: Then, because the points E, C are in this plane, the ftraight line EC is in it a: for the same reafon, the ftraight line BC is in the fame; and, by the hypothefis, EB is in it therefore the three ftraight lines EC, CB, BE are in one plane: but the whole of the lines DC, AB, E a 5.def. r. B C and BC produced, are in the fame plane with the parts of them |