1. WHAT is meant by a solid in geometry? What are the boundaries of solids? How many dimensions has a solid ? 2. Explain the distinction between a plane surface and a curved surface. 3. What is assumed in speaking of a plane? Three points are requisite to fix the position of a plane. Is there any exception to this proposition? 4. Shew that every two points are in the same straight line, and every three are in the same plane. 5. How is the inclination of a straight line to a plane measured? 6. How many straight lines can be drawn making a given angle, (1) with a straight line, (2) with a plane. Shew that if the given angle be a right angle, there is only one such straight line. 7. What is meant by the projection of a straight line on a plane? 8. State what is to be considered the inclination to each other of two straight lines in space, which do not meet when produced. 9. Define the inclination of a plane to a plane, and shew that it is the same at all points of their intersection. 10. Two planes are parallel to each other when they are equidistant, or when all the perpendiculars that can be drawn between them are equal. 11. When is a straight line perpendicular to a plane? Shew that it is so when it is perpendicular to two lines in that plane. 12. How must one plane meet another, so that the inclination of the planes may be equal to a given angle? 13. Three straight lines which meet in a point, and are perpendicular to a fourth straight line, are in the same plane. If they meet, but not in one point, are they in the same plane? 14. If a plane be defined as the surface generated by the revolution of a straight line, which is always perpendicular to a given straight line, and passes through a given point in it; shew that the straight line joining any two points in a plane will be wholly in that plane. 15. Can any reason be assigned, why the same order has not been followed in Euc. XI, 8, 9, as in Euc. 1, 11, 12? 16. Define a solid angle, and shew in how many ways a solid angle may be formed with equilateral triangles and squares. 17. Can a solid angle be formed with any three plane angles assumed at pleasure? 18. How is a solid angle measured? 19. What is the limit of the sum of the plane angles which together can form a solid angle? 20. Can it be justly said that the parallelopiped and the cube have the same relation to each other as the rectangle and the square? 21. What is the length of an edge of a cube whose volume shall be double that of another cube whose edge is known? 22. If a straight line be divided into two parts, the cube on the whole line is equal to the cubes on the two parts together with thrice the right parallelopiped contained by their rectangle and the whole line. 23. When a cube is cut by a plane obliquely to any of its sides, the section will be a rectangular parallelogram, always greater than a side of the cube, if made by cutting the opposite sides. 24. Shew how to draw a plane cutting two adjacent sides of a cube, so that the section shall be equal and similar to a side of the cube. 25. The content of a regular parallelopipidon whose length is any multiple of the breadth, and breadth the same multiple of the depth, is the same as that of a cube whose edge is the breadth. 26. If a, b, c be the three dimensions, and v the volume of a parallelo2 {(a+b) v + a2 b2} * piped, prove that the superficies is equal to ab 27. How is it shown that the cube described with a given line as one of the edges, is eight times the cube described with half the line as one of its edges? 28. Shew how to transform a given cube into a parallelopiped, whose three adjacent edges shall be in continual proportion. 29. Is every possible section of a parallelopiped which can be made, a parallelogram? 30. Shew how to bisect a parallelopiped, so that the area of the section may be the greatest possible. 31. There are two cylinders of equal altitudes, but the base of one of them is three times that of the other: compare the volumes of the cylinders. 32. How is a right cone generated? What is meant by the axis and by the base of a cone? 33. What is Euclid's definition of similar solid figures contained by planes? Is this definition liable to any objection? 34. Shew how a prism, pyramid, cylinder and cone may be generated. In what respects does a prism differ from a pyramid ? 35. Shew how a triangular prism may be divided into three equal triangular pyramids of the same base and altitude: and find into how many triangular pyramids a prism can be divided, the base of which is a polygon of n sides. 36. Shew how to find the content of a pyramid, whatever be the figure of the base, the altitude and area of the base being given. 37. What solid figure is that, which if cut in any direction whatever by planes, the sections shall be similar? 38. If two triangular prisms have the same base and equal ends, they cannot have their upper edges not coincident. 39. What will be the form of the base of a pyramid whose sides consist of the greatest possible number of equilateral triangles ? 40. Having given six straight lines of which each is less than the sum of any two; determine how many tetrahedrons can be formed, of which these straight lines are the edges. 41. Why cannot a sheet of paper be made to represent the vertex of a pyramid, without folding? 42. Define the generation of a sphere. Can any reason be assigned why Euclid has not defined a circle in a similar manner, as the figure generated in a plane by the revolution of a straight line about one of its extremities which remain fixed? 43. Shew that the ratio of the diameter of a sphere, and the side of the inscribed cube, is as three to unity. 44. Mention the names and define the five regular solids. GEOMETRICAL EXERCISES ON BOOK XI. THEOREM I. Prove that if a straight line be perpendicular to a plane, its projection on any other piane, produced if necessary, will cut the common intersection of the two planes at right angles. Let AB be any plane and CEF another plane intersecting the former at any angle in the line EF; and let the line GH be perpendicular to the plane CEF. Draw GK, HL perpendicular on the plane AB, and join LK, then LK is the projection of the line GH on the plane AB; produce EF, to meet KL in the point L; then EF, the intersection of the two planes, is perpendicular to LK, the projection of the line GH on the plane AB. Because the line GH is perpendicular to the plane CEF, every plane passing through GH, and therefore the projecting plane GHKL is perpendicular to the plane CEF; but the projecting plane GHLK is perpendicular to the plane AB; (constr.) hence the planes CEF, and AB are each perpendicular to the third plane GHLK; therefore EF, the intersection of the planes AB, CEF, is perpendicular to that plane; and consequently, EF is perpendicular to every straight line which meets it in that plane.; but EF meets LK in that plane. Wherefore, EF is perpendicular to KL, the projection of GĦ on the plane AB. THEOREM II. Prove that four times the square described upon the diagonal of a rectangular parallelopiped, is equal to the sum of the squares described on the diagonals of the parallelograms containing the parallelopiped. Let AD be any rectangular parallelopiped; and AD, BG two diagonals intersecting one another; also AG, BD, the diagonals of the two opposite faces HF, CE. Then it may be shewn that the diagonals AD, BG, are equal; as also the diagonals which join CF and HE: and that the four diagonals of the parallelopiped are equal to one another. The diagonals AG, BD of the two opposite faces HF, CE are equal to one another: also the diagonals of the remaining pairs of the opposite faces are respectively equal. And since AB is perpendicular to the plane CE, it is perpendicular to every straight line which meets it in that plane, therefore AB is perpendicular to BD, and consequently ABD is a right-angled triangle. And the square on AD is equal to the squares on AB, BD, (1. 47.) also the square on BD is equal to the squares on BC, CD, therefore the square on AD is equal to the squares on AB, BC, CD; similarly the square on BG or on AD is equal to the squares on AB, BC, CD. Wherefore the squares on AD and BG, or twice the square on AD, is equal to the squares on AB, BC, CD, AB, BC, CD; but the squares on BC, CD are equal to the square on BD, the diagonal of the face CE; similarly, the squares on AB, BC are equal to the square on the diagonal of the face HB: also the squares on AB, CD, are equal to the square on the diagonal of the face BF; for CD is equal to BE. Hence, doublethe square on AD is equal to the sum of the squares on the diagonals of the three faces HF, HB, BC. In a similar manner, it may be shewn, that double the square on the diagonal is equal to the sums of the squares on the diagonals of the three faces opposite to HF, HB, BC. Wherefore, four times the square on the diagonal of the parallelopiped, is equal to the sum of the squares on the diagonals of the six faces. I. 3. If two straight lines are parallel, the common section of any two planes passing through them is parallel to either. 4. If two straight lines be parallel, and one of them be inclined at any angle to a plane; the other also shall be inclined at the same angle to the same plane. 5. If two straight lines in space be parallel, their projections on any plane will be parallel. 6. Shew that if two planes which are not parallel be cut by two other parallel planes, the lines of section of the first by the last two will contain equal angles. 7. If four straight lines in two parallel planes be drawn, two from one point and two from another, and making equal angles with another plane perpendicular to these two, then if the first and third be parallel, the second and fourth will be likewise. 8. Draw a plane through a given straight line parallel to another given straight line. 9. Through a given point it is required to draw a plane parallel to both of two straight lines which do not intersect. 10. From a point above a plane two straight lines are drawn, the one at right angles to the plane, the other at right angles to a given line in that plane; shew that the straight line joining the feet of the perpendiculars is at right angles to the given line. 11. AB, AC, AD are three given straight lines at right angles to one another, AE is drawn perpendicular to CD, and BE is joined. Shew that BE is perpendicular to CD. 12. Two planes intersect each other, and from any point in one of them a line is drawn perpendicular to the other, and also another line perpendicular to the line of intersection of both; shew that the plane which passes through these two lines is perpendicular to the line of intersection of the planes. 13. Find the distance of a given point from a given line in space. 14. Draw a line perpendicular to two lines which are not in the same plane. 15. Two planes being given perpendicular to each other, draw a third perpendicular to both. 16. Two perpendiculars are let fall from any point on two given planes, shew that the angle between the perpendiculars will be equal to the angle of inclination of the planes to one another. 17. Two planes intersect, straight lines are drawn in one of the planes from a point in their common intersection making equal angles with it, shew that they are equally inclined to the other plane. II. 18. Three straight lines not in the same plane, but parallel to and equidistant from each other, are intersected by a plane, and the points |