I. A SOLID is that which hath length, breadth, and thickness. II. That which bounds a folid is a fuperficies. III. A straight line is perpendicular, or at right angles, to a plane, when it makes right angles with every straight line meeting it in that plane. IV. A plane is perpendicular to a plane, when the straight lines drawn in one of the planes perpendicularly to the common section of the two planes, are perpendicular to the other plane. V. 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. VI. The inclination of a plane to a plane is the acute angle contained by two ftraight 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. VII. Two planes are faid to have the same, or a like inclination to one another, which two other planes have, when the faid angles of inclination are equal to one another. VIII. Parallel planes are fuch which do not meet one another tho' produced. IX. Book XI. A solid angle is that which is made by the meeting of more than See N. two plane angles, which are not in the fame plane, in one point. X. The tenth Definition is omitted for reasons given in the Notes.' See N. XI. Similar folid figures are fuch as have all their folid angles equal, See N. each to each, and which are contained by the same number of fimilar planes. XII. A Pyramid is a folid figure contained by planes that are constituted betwixt one plane and one point above it in which they meet. XIII. A Prism 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. XIV. A Sphere is a folid figure described by the revolution of a femicircle about its diameter, which remains unmoved. XV. The axis of a sphere is the fixed straight line about which the semicircle revolves. XVI. The center of a sphere is the fame with that of the femicircle. XVII. The diameter of a sphere is any ftraight line which passes thro' the center, and is terminated both ways by the fuperficies of the fphere. 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. Book XI. If the fixed fide be equal to the other fide containing the right angle, the Cone is called a right angled Cone; if it be lefs than the other fide, an obtuse angled, and if greater, an acute angled Cone. XIX. The axis of a Cone is the fixed ftraight line about which the triangle revolves. XX. The bafe of a Cone is the circle defcribed by that fide containing the right angle, which revolves. XXI. A Cylinder is a folid figure described by the revolution of a right angled parallelogram about one of its fides which remains fixed. XXII. The axis of a cylinder is the fixed ftraight line about which the parallelogram revolves. XXIII. The bases of a cylinder are the circles defcribed by the two revolving oppofite fides of the parallelogram. XXIV. Similar cones and cylinders are those which have their axis and the diameters of their bafes proportionals. XXV. A Cube is a folid figure contained by fix equal fquares. XXVI. A Tetrahedron is a folid figure contained by four equal and equilateral triangles. XXVII. An Octahedron is a folid figure contained by eight equal and equilateral triangles. XXVIII. A Dodecahedron is a folid figure contained by twelve equal pentagons which are equilateral and equiangular. XXIX. An Icofahedron is a folid figure contained by twenty equal and equilateral triangles. DEF. A. A Parallelepiped is a folid figure contained by fix quadrilateral figures whereof every oppofite two are parallel. PROP. I. THEOR. Book XI. NE part of a straight line cannot be in a plane and See N. another part above it. OND If it be poffible, let AB part of the straight line ABC be in the plane, and the part BC above it. and since 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 pafs thro' the ftraight line AD, and be turned about it until it pass thro' the point C; and because the points B, C are in this plane, the A B D ftraight line BC is in it a. therefore there are two straight lines a. 7. Def. I. ABC, ABD in the fame plane that have a common fegment AB, which is impoffible. Therefore one part, &c. Q. E. D. PROP. II. THEOR. TWO lines b.Cor.11.1. WO straight lines which cut one another are in See N. one plane, and three ftraight lines which meet one another are in one plane. Let two ftraight lines AB, CD cut one another in E; AB, CD are in one plane. and three straight lines EC, CB, BE which meet one another, are in one plane." Let any plane pass thro' the straight line A EB, and let the plane be turned about EB, produced if neceffary, until it pass thro' the point C. then because the points E, C are in this plane, the straight line EC is in it 2. for the fame reason, the straight line BC is in the fame; and, by the Hypothesis, EB is in it., therefore the three ftraight lines EC, · CB, BE are in one plane. but in the plane C in which EC, EB are, in the fame are ↳ CD, E D B AB. therefore AB, CD are in one plane. Wherefore two straight lines, &c. Q. E. D. 2. 7. Def. z. b. I. II. Book XI. See N. IF PROP. III. THEOR. F two planes cut one another, their common fection is a straight line. Let two planes AB, BC cut one another, and let the line DB be their common fection; DB is a straight line. If it be not, from the point D to B B draw in the plane AB the ftraight line E line DFB. then two ftraight lines DEB, F D A a. 10. Ax.I. is impoffible a. therefore BD the common See N. fection of the planes AB, BC cannot but be a straight line. Wherefore if two planes, &c. Q. E. D. I PROP. IV. THEOR. TF a ftraight line ftand at right angles to each of two ftraight lines in the point of their interfection, it fhall also be at right angles to the plane which passes through them, that is, to the plane in which they are. Let the ftraight line EF ftand at right angles to each of the ftraight lines AB, CD in E the point of their interfection. EF is also at right angles to the plane paffing thro' AB, CD. Take the straight lines AE, EB, CE, ED all equal to one another; and thro' E draw, in the plane in which are AB, CD, any ftraight line GEH; and join AD, CB; then from any point F in EF, draw FA, FG, FD, FC, FH, FB. and because the two straight lines AE, ED are equal to the two BE, EC, and that they contain a. 15. 1. equal anglesa AED, BEC, the base AD is equal b to the base BC, b. 4. I. and the angle DAE to the angle EBC. and the angle AEG is equal to the angle BEHa; therefore the triangles AEG, BEH have two angles of one equal to two angles of the other, each to each, and the fides AE, EB, adjacent to the equal angles, equal to one another; c. 26. I. wherefore they fhall have their other fides equal . GE is therefore |