CB, describe the semicircle DBE, and from AB cut off AF AD; then since AB is perpendicular to CB, it is a tangent to the circle at B; .. (Prop. 67), EA; AB=AB: AD; but since DE and AB are each double of CB, they are equal, and AF AD; .. (Alg. 107), EA-AB: AB= AB-AD: AD, which, from the above mentioned equalities, gives AF: AB=BF: AF; hence, (Alg. 105), AB: AF =AF: BF. Cor. Since AB-DE, the first proportion gives AE: ED =ED: AD; .. the line AE is also divided in the same manner. SCHOLIUM. A line divided as in the proposition, is said to be divided in extreme and mean ratio, and also to be cut in medial section, or to be divided medially. GEOMETRY OF PLANES. DEFINITIONS.-I. A solid is that which hath length, breadth, and thickness. II. The boundaries of a solid are superficies. III. A straight line is perpendicular, or at right angles to a plane, when it makes right angles with every line which meets it in that plane. IV. A plane is perpendicular to a 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. 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 in the first line, meets the same plane. VI. The inclination of a plane to a plane is the acute angle contained by two straight lines drawn from any the same point of their common section, at right angles to it, one upon one plane, and the other upon the other plane. VII. Two planes are said to have the same, or like inclinations to one another, which two other planes have, when their angles of inclination are equal to one another. VIII. Parallel planes are such as do not meet one another though produced. IX. A straight line and a plane are parallel, if they do not meet when produced. X. The angle formed by two intersecting planes is called a dihedral angle, and is measured as in Def. 6. XI. Any two angles are said to be of the same affection, when they are either both greater or both not greater than a right angle. The same term is applied to arcs of the same or equal circles, when they are either both greater or both not greater than a quadrant. PROPOSITION LXXVIII.-THEOREM. Any three straight lines which meet one another, not in the same point, are in one plane. Let the straight lines AB, BC, D E B Let any plane pass through the straight line AB, and let the plane, produced if necessary, be turned about AB, till it pass through the point C. Then, because the points B and C are in the plane, the straight line BC is in the plane, (Def. 7); and because the points C and E are in the plane, the CE or CD is in the plane, and by hypothesis AB is in the plane; .. the three straight lines AB, BC, CD are all in one plane. Cor. 1. Any two straight lines that cut one another are in one plane. Cor. 2. Only one plane can pass through three points, or through a straight line and a point. Cor. 3. Any three points are in one plane. PROPOSITION LXXIX.-Theorem. If two planes cut one another, their common section is a straight line. Let two planes AB, BC cut one another, and let B and D be two points in the line of their common section. From B to D draw the straight line BD, then since B and D are points in the plane AB, the line BD is in that plane, (Def. 7); for the same reason it is in the plane CB; .. being in each of the planes, it is their common section; hence the common section of the two planes is a straight line. PROPOSITION LXXX.-THEOREM. If a straight line stand at right angles to each of two straight lines in the point of their intersection, it will also be at right angles to the plane in which these lines are. Let PO be to the lines AB, CD, at their point of intersection O, it is to their plane. For take OB=OD, and join PB, PD, and BD, and draw any line FE, meeting BD in E, and join PE. B E Then BO=DO, and PO common to the two As POB, POD, and the contained angles Ls, the base PB= PD, .. the PBD is isosceles, and the OBD is isosceles by construction. Now. POD is a L, PO2+OD2=PD2, but OD2-OE2+DE-EB, (Prop. 44), and PD2=PE2+ DE EB, .. PO2+OE2+DE·EB= PE2+ DE EB; take the rectangle DE EB from both, and there remains PO2+ OE=PE,.. the LPOE is a right angle, and PO is at Ls to EO; in the same manner, if the line EO had been in any of the other angles BOC, COA, or AOD, it could be demonstrated to be at rLs to PO; . when PO is at rs to two straight lines at the point of their intersection, it is at right angles to every line in that plane. Cor. 1. If a plane be horizontal in any two directions, it is so in every direction. Cor. 2. The perpendicular PO is less than any oblique line, as PB, (Prop. 23), and therefore the perpendicular measures the shortest distance from the point P to the plane. PROPOSITION LXXXI.-THEOREM. If three straight lines meet all in one point, and a straight line stand at right angles to each of them in that point, these three straight lines are in one and the same plane. Let the AB stand at t Ls to each I of the is BC, BD, BE, in B, the point where they meet; BC, BD, and BE are in one and the same plane. If not, let, if possible, two of them, as BD, BE, be in the same plane, and BC above it, and let a plane pass B through AB, BC, and cut the plane in which BD and BE are in the BF, (Prop. 79); then since AB is at o Ls to each of the Is BD, BE, at the point of their intersection, it is also at Ls to BF, (Prop. 80), which is in the same plane; .. the ABF is a r; but by hypothesis, the LABC is also a ; hence the LABF= the LABC, and they are both in the same plane, which is impossible; .. the | BC is not above the plane in which are BD and BE, and in the same manner it may be shown that it is not below it. Wherefore the three s BC, BD, BE, are in one and the same plane. PROPOSITION LXXXII.—THEOREM. If two straight lines be at right angles to the same plane, they shall be parallel to one another. Let the Is AB, CD be at t Ls to the same plane; AB is || CD. Let them meet the plane in the points B, D, and draw the | BD, to which draw DE at Ls in the same B plane; and make DE=AB, and join BE, AE, AD. Then AB is to the plane, it shall make Ls with E every which meets it and is in that plane, (Def. 3); but BD, BE, which are in that plane, do each of them meet AB;. each of the Ls ABD, ABE is a L. For the same reason each of the Ls CDB, CDE is a r1L; and ". AB-DE, and BD common, the two sides AB, BD are the two ED, DB, and they contain t Ls. ..the base AD BE, (Prop. 5). Again, AB-DE, and BEAD, and the base AE common to the As ABE, EDA, the LABE the LEDA, (Prop. 9); but ABE is a 'L, .. EDA is also a rL, and ED is DA; but it is also to each of the two BD, DC: wherefore ED is at Ls to each of the three Is BD, DA, DC, in the point in which they meet, these three s are all in the same plane, (Prop. 81); but AB is in the plane in which are BD, DA, (Prop. 78), since any three Is which meet one another are in one plane. AB, BD, DC are in one plane; and each of the Ls ABD, BDC is a ; hence (Prop. 16, cor. 2) AB is || CD. PROPOSITION LXXXIII-THEOREM. If two straight lines be parallel, and one of them is at right angles to a plane, the other shall also be at right angles to the same plane. to which AB is, let DG be to it. Then (Prop. 82) DG is || AB; .. DG and DC are both || AB, and are drawn through the same point D, which is impossible. PROPOSITION LXXXIV.-THEOREM. I If two straight lines be each of them parallel to the same straight line, though not both in the same plane with it, they are parallel to one another. Let AB; CD be each of them EF, and not in the same plane with it; AB shall be || CD. C E 7. H K D And be to the AB; .. In EF take any point G, from which draw, in the plane passing through EF, AB, the GH at Ls to EF; and in the plane passing through EF, CD, draw GK at Ls to the same EF. cause EF is both to GH and GK, EF is plane HGK, passing through them; and EF is || AB is at Ls to the plane HGK, (Prop. 83). For the same reason, CD is likewise at Ls to the plane HGK. Hence AB, CD are each of them at Ls to the plane HGK. . (Prop. 82) AB || CD. PROPOSITION LXXXV.-THEOREM. If two straight lines AB, BC, meeting one another, be parallel to two others DE, EF, B that meet one another, though not in the same plane with the first two, the first two and the other two shall contain equal angles. E D Take BA, BC, ED, EF, all equal to one another; and join AD, CF, BE, AC, DF; BA is and || ED, AD is both and || (Prop. 24, cor. 1) to BE; for the same reason CF is and || BE; .:. AD and CF are each of them and || BE. But Is that are || the same are one another, (Prop. 84); . AD is || CF; and it is equal to it, and AC, DF, join them towards |