71. If two lines be drawn from any point without a circle to intersect it, and lines be drawn to the alternate points of intersection, these will always intersect in the chord which joins the points of contact of the tangents drawn to the circle from the point without it. 72. If the radius of a circle be divided in extreme and mean ratio, the greater segment is the side of the regular decagon inscribed in that circle; and the sum of the squares of the radius and its greater segment is equal to the square of the side of the inscribed regular pentagon. 73. If a tangent be drawn to a circle equal to its diameter, and from the extremity of the tangent a line be drawn through the centre, and from the points of intersection of this line with the circle, lines be drawn to the point of contact: the greater of these will be the radius of a circle in which the less will be the side of the inscribed decagon, and in which the tangent will be the side of the inscribed pentagon. OF PLANES AND SOLIDS. THE figures which we have hitherto considered are such as lie entirely on one plane in those which follow, the intersections of different planes with one another, or with given straight lines, the volumes of space enclosed with certain combinations of planes, and other topics of the same kind, are the objects of research. The conception of the figures and of their properties is greatly facilitated by the use of models, and no student should proceed without them; though, of course, no great regard to extreme precision is requisite in their con struction. DEFINITIONS." 1. (88.) The common section of two planes is the line in which they meet or cut each other. 2. (89.) A line is perpendicular to a plane, when it is perpendicular to every line in that plane which meets it; and the point in which the perpendicular meets the plane is called the foot of the perpendicular. 3. (90.) One plane is perpendicular to another plane, when every line of the one, which is perpendicular to the line of their common section, is perpendicular to the other. 4. (91.) The inclination of one plane to another, or the angle they form between them, is the angle contained by two lines, drawn from any point in the common section and at right angles to it, one of these lines in each plane. This is often called a dihedral angle. 5. (91a.) If from a point in a line which meets a plane, a perpendicular be drawn to the plane, and the points of intersection of these two lines with the plane be joined, the angle formed by the line in the plane and the line which meets the plane is called the inclination of the line to the plane. 6. (92.) Parallel planes are such as being produced ever so far in every direction will never meet. 7. (93.) A solid angle is that which is made by three or more plane angles meeting each other in the same point. * A modified arrangement of the definitions and propositions of this subject has rendered it necessary to commence both as with a new subject. The numbers of the last edition, however, for obvious reasons, being desirable to be retained, they are here marked in parentheses. 8. (94.) Similar solids, contained by plane figures, are such as have all their solid angles equal, each to each, and are bounded by the same number of similar planes, and placed in the same consecutive order. 9. (95.) A prism is a solid whose ends are parallel, equal, and like plane figures; and its sides, connecting those ends, are parallelograms. 10. (96). A prism takes particular names according to the figure of its base or 'ends, whether triangular prism, square prism, rectangular prism, pentagonal prism, hexagonal prism, and so on. 11. (97.) A right prism is that which has the planes of the sides perpendicular to the planes of the ends or base. 12. (98.) A parallelopiped, or parallelopipedon, is a prism bounded by six parallelograms, every opposite two of which are equal and parallel. 13. (99.) A rectangular parallelopipedon is that whose bounding planes are all rectangles. 14. (100.) A cube is a square prism, being bounded by six equal square sides or faces. 15. (101.) A cylinder is a round prism, having circles for its ends. It is conceived to be formed by the rotation of a right line about the circumferences of two equal and parallel circles, always parallel to the axis. 16. (102.) The axis of a cylinder is the right line joining the centres of the two parallel circles about which the figure is described. When the axis of the cylinder is at right angles to the planes of the parallel ends, the cylinder is called a right, and when oblique to them an oblique cylinder. 17. (103.) A pyramid is a solid, whose base is any right lined plane figure, and its sides triangles, having all their vertices meeting together in a point without the plane of the base, called the vertex of the pyramid. 18. (104.) A pyramid, like the prism, takes its particular name from the figure of the base; as a triangular, quadrangular, etc. pyramid. 19. (105.) A cone is a round pyramid, having a circular base. It is conceived to be generated by the rotation of a right line about the circumference of a circle, one end of which is fixed at a point without the plane of that circle. 20. (106.) The axis of a cone is the right line joining the vertex, or fixed point, and the centre of the circle about which the figure is described. When the axis of the cone is at right angles to its base, the cone is said to be a right, and when oblique to the base, an oblique cone. 21. (107.) Similar cones and similar cylinders are such as have their altitudes, the diameters of their bases, and their axes, proportional. 22. (108.) A sphere is a solid bounded by one curve surface, which is every where equally distant from a certain point called the centre. It is sometimes conceived to be generated by the rotation of a semicircle about its diameter, which remains fixed. 23. (109.) The axis of a sphere is the right line about which the semicircle revolves, and the centre is the same as that of the revolving semicircle. 24. (110.) The diameter of a sphere is any right line passing through the centre, and terminated both ways by the surface. 25. (111.) The altitude of a solid is the perpendicular drawn from the vertex to the opposite face, considered as its base. 26.' (112.) By the distance of a point from a plane is meant the shortest line that can be drawn from that point to meet the plane. It is subsequently shown that this is the perpendicular (th. 3, cor. 1). 349 THEOREMS. SECT. I.-OF LINES AND PLANES. THEOREM I. (96.) Two straight lines which meet each other; the three sides of a triangle; any three points in space; or two parallel lines;—are in the same plane, and being given determine its position. First. LET AB, AC, be two straight lines which intersect each other in A. A plane may be made to pass through AB in any direction, and hence it may be turned about AB till it also passes through C. Then the line AC which has two of its points, A and C, in this plane, lies wholly in the plane, and the plane itself is fixed in its position. D E Second. A triangle ABC, or any three points in space not in the same right line, determine the position of a plane. Third. Also two parallels, AB, CD, determine the position of the plane in which they are situated. For the plane may be turned about one of them to touch a point of the other, and the second line being in the same plane as the first, and passing through a point in it, the plane must be that just determined. THEOREM II. (97.) The common section of two planes is a right line. (Same figure.) LET ACBDA, AEBFA, be two planes cutting each other, and A, B, two points in which the two planes meet; drawing the line AB, this line will be the common intersection of the two planes. For, because the right line AB touches each of the planes in the points A and B, it touches them in all other points (def. 20): this line is therefore common to the two planes. That is, the common intersection of the two planes is a right line. THEOREM III. (98.) If a straight line be perpendicular to any two other straight lines in their point of intersection, it shall also be at right angles to the plane which passes through them, that is, to the plane in which they are. LET PC, PB, be two lines intersecting in P, and AP be another line passing through P at right angles to PB and PC, then AP will be perpendicular to any line PQ in the plane BPC. For, in PQ take any point Q, and draw QR parallel to PC, meeting PB in R; and take RB equal to RP, and draw BQ to meet PC in C. Join AC, AQ, AB. Then, since the line QR is drawn M N parallel to the side PC of the triangle PCB, we have (th. 82) PR : RB :: CQ : QB, and PR = RB; hence BC is bisected in Q. Then, (th. 38,) PC2 + PB2 = 2PQ2 + 2CQ2, and AC2 + AB2 = 2AQ2 + 2CQ2; and taking the first equation from the second, we shall have (th. 34) AP2 + AP2 = 2AQ2 2PQ3, or AQ2 = AP2 + PQ2. Whence (1 cor. th. 34) APQ is right angled at P, or AP is perpendicular to PQ. And the same may be proved for any other line drawn in the plane MN through the point P. The line AP is, therefore, perpendicular to every straight line in the plane MN passing through P; and hence to the plane in which those lines are (def. 2). Cor. 1. The perpendicular AP is the shortest line that can be drawn from A to the plane. See def. 26. Cor. 2. Oblique lines which meet the plane at the same distance from the foot of the perpendicular and proceed from the same point in the perpendicular, are equal to one another: and that which meets the plane at a less distance from P is less than that which meets it at a more remote distance. THEOREM IV. (99.) There can only be one line perpendicular to a given plane, and passing through a given point, whether that point be in the plane or without it. FOR suppose there can be two. First. Let P be in the plane HK, and the two perpendiculars be PQ, PR. Through QPR let a plane pass, cutting HK in PN. Then the angles QPN, RPN will both be right angles (def. 89), and hence equal to one another (ax. 10): that is, a part equal to the whole, which is absurd. Hence PR is perpendicular to the plane HK. Second. Let P be without the plane HK, and let PQ, PR be the two perpendiculars admitted for the moment to be drawn from P to HK. Let the plane MN passing PQ, and PR, cut the plane HK in QR. Then the angles PQR, PRQ are both right angles, which is impossible (th. 17). Hence, there cannot be two perpendiculars drawn to the same plane from a point without it. THEOREM V. (100.) If a straight line be perpendicular to one of two parallel planes, it will be perpendicular to the other. LET HK, LM, be two parallel planes, and AB be perpendicular to HK, it shall also be perpendicular to LM. For if not, from A draw AC perpendicular to LM, meeting it in C, and through ABC draw a plane cutting the planes HK, LM, in AD and BCE. Then, since AC is perpendicular to LM, the angle ACB is a right angle, and hence ABC is HA L D K N M less than a right angle. Hence, since BAD is a right angle, the two angles DAB and ABC are less than two right angles, and hence the lines AD, BE, in the same plane will meet if sufficiently produced. But AD is in the plane HK, and BC in LM, hence HK also meets LM: which is impossible, since by hypothesis they are parallel. THEOREM VI. (101.) [See figure to theorem 5]. If two planes be perpendicular to the same straight line, they are parallel to one another. LET the planes HK and LM be perpendicular to the line AB, they will be parallel to one another. For if they be not parallel they must meet. Let N be a point in their common intersection and join NA, NB. Then since AB is perpendicular to the plane HK, it is perpendicular to NA drawn through A in that plane, and NAB is a right angle. In like manner, NBA is a right angle. But NAB being a triangle, the two angles NAB, NBA, are together less than two right angles. Hence the perpendiculars from A, B, in the planes HK and LM, do not meet at N. In the same manner it can be proved that they do not meet at any other point; and hence that the planes HK, LM, have not any point common, and are therefore parallel. THEOREM VII. (102.) If from the foot of the perpendicular to any plane a line be drawn at right angles to a line in that plane, any line drawn from the point of intersection to a point in the line which is perpendicular to the plane will also be perpendicular to the line which lies in the plane. LET AP be perpendicular to the plane MN, and BC be a line situated in that plane: if from P, the foot of the perpendicular, the line PQ be drawn perpendicular to BC, then QA drawn to any point A, in AP, will also be perpendicular to BC. Take BQ = QC, and join PB, PC, AB, AC. Then since BQ = QC, and PQB PQC, and PQ common, we have also PB PC. Again, since PB PC, APB APC (def. 89), and AP common, we have also AB AC. Then AQ, QB, being equal to AQ, QC, each to each, and AB equal to AC, the angle AQB = AQC, and hence they are right angles. Cor. BC is perpendicular to the plane APQ, for BC is perpendicular to AQ and PQ, which determine that plane (th. 1). |