The axis of a cone is the fixed ftraight line about which the triangle revolves. XX. The base of a cone is the circle described by that fide containing the right angle, which revolves. XXI. A cylinder is a folid figure defcribed 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 straight line about which the parallelogram revolves. XXIII. The bafes of a cylinder are the circles described by the two re volving oppofite fides of the parallelogram. XXIV. Similar cones and cylinders are those which have their axes and the diameters of their bafes proportionals. XXV. A cube is a folid figure contained by fix equal fquares. 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 parallelopiped is a folid figure contained by fix quadrilateral figures whereof every oppofite two are parallel. PROP. ON PROP. I. THEOR. Book XI. NE part of a ftraight line cannot be in a plane and See N. another part above it. If it be poffible, let AB, part of the ftraight 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: And let any plane C points B, C are in this plane, the straight line BC is in it: a 7. def. 1. Therefore there are two ftraight lines ABC, ABD in the fame plane that have a common fegment AB, which is impoffible b. b Cor. 11.1. Therefore one part, &c. Q. E. D. TW WO straight lines which cut one another are in one See N. 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 one plane: And three ftraight lines EC, CB, BE which met one another, are in one plane. Let any plane pass through the straight line EB, and let the plane be turned about EB, produced, if neceffary, until it pass through the point C: Then because the points E, C are in this plane, the ftraight line EC is in it: For the fame reason, 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 in the plane in which EC, EB are, in the fame are b CD, AB: Therefore AB, C) are in E one plane. Wherefore two ftraight lines, &c. Q. E. D. a 7. def. I. Book XI. PROP. III. THEOR. Sel N. IF two planes cut one another, their common fection is a ftraight line. B E Let two planes AB, BC cut one another, and let the line DB be their common fection; DB is a ftraight line: If it be not, from the point. D to B, draw, in the plane AB, the ftraight line DEB, and in the plane BC the ftraight line DFB: Then two straight lines DEB, DFB have the fame extremities, and therefore include a space bea 10. Ax. 1. twixt them; which is impoffible: Therefore BD the common section of the planes AB, BC cannot but be a straight line. Wherefore, if two planes, &c. Q, E. D. C A PRO P. IV. THEO R. IF a ftraight line ftand at right angles to each of two ftraight lines in the point of their interfection, it shall alfo be at right angles to the plane which pafles through them, that is, to the plane in which they are. Let the ftraight line EF ftand at right angles to each of the traight lines AB, CD in E, the point of their interfection: EF is alfo at right angles to the plane paffing through AB, CD. a Take the ftraight 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 GFH; and join AD, CB; then, from any point F in EF, draw FA, FG, FD, FC, FH, FB: And becaufe the two ftraight lines AE, ED are equal to the two BE, EC, and that they contain equal angles AED, BEC, the bafe AD is equal to the bafe BC, and the angle DAE to the angle EBC: And the angle AEG is equal to the angle BEH; 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; wherefore they fhall have their other fides equal: GE is therefore equal b 4. I. F d 8. 1. equal to EH, and AG to BH: And because AE is equal to EB, Book XI. and FE common and at right angles to them, the base AF is equal to the bafe FB; for the fame reason, CF is equal to FD: And because AD is equal to BC, and AF to FB, the two fides FA, AD are equal to the two FB, BC, each to each; and the bafe DF was proved equal to the bafe FC; therefore the angle FAD is equal to the angle FBC: Again, it was proved that GA is equal to BH, and alfo AF A to FB; FA then and AG, are equal to FB and BH, and the angle FAG has been proved equal to the angle FBH; therefore the bafe GF is equal to the bafe FH: Again, because it was proved, that GE is equal to EH, and EF is common; GE, EF are equal to HE, EF; and the bafe GF C E H B is equal to the base FH; therefore the angle GEF is equal to the angle HEF; and confequently each of these angles is a right angle. Therefore FE makes right angles with GH,e ro def. 1. that is, with any ftraight line drawn through E in the plane paffing through AB, CD. In like manner, it may be proved, that FE makes right angles with every straight line which meets it in that plane. But a ftraight line is at right angles to a plane when it makes right angles with every straight line which meets. it in that plane f: Therefore EF is at right angles to the plane f 3. def. II. in which are AB, CD. Wherefore, if a straight line, &c. QE. D. PROP. V. THEO R. IF three ftraight lines meet all in one point, and a se N. ftraight line ftands at right angles to each of them in that point; these three straight lines are in one and the fame plane. Let the ftraight line AB ftand at right angles to each of the ftraight lines BC, BD, BE, in B the point where they meet; BC, BD, BE are in one and the fame plane. If not, let, if it be poffible, BD and BE be in one plane, and BC be above it; and let a plane pass through AB, BC, the common fection of which with the plane, in which BD and BE are, a 3. II. a Book XI. are, fhall be a straight line; let this be BF: Therefore the three ftraight lines AB, BC, BF are all in one plane, viz. that which paffes through AB, BC: And because AB ftands at right angles to each of the ftraight lines BD, BE, it is also at right angles b to the plane paffing through them; and therefore makes €3. def. 11. right angles with every ftraight A. b 4. II. line meeting it in that plane; but line BC is not above the plane in C F D E which are BD and BE: Wherefore the three ftraight lines BC, BD, BE are in one and the fame plane. Therefore, if the ftraight lines, &c. Q. E, D. b 4. def. II. F two ftraight lines be at right angles to the fame plane, they fhall be parallel to one another. Let the ftraight lines AB, CD be at right angles to the fame plane; AB is parallel to CD. Let them meet the plane in the points B, D, and draw the ftraight line BD, to which draw DE at right angles, in the fame plane; and make DE equal to AB, and join BE, AE, AD. Then, becaufe E D ED, DB; and they contain right angles; therefore the base to |