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 + a2b2} ̧ 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 plane, 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 ĺ;

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 GHK

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 GH 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.
Similarly, GDB 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

of intersection joined; shew when the triangle thus formed will be equilateral and when isosceles.

19. Three straight lines, not in the same plane, intersect in a point, and through their point of intersection another straight line is drawn within the solid angle formed by them; prove that the angles which this straight line makes with the first three are together less than the sum, but greater than half the sum, of the angles which the first three make with each other.

20. If two solid angles bounded by any number of plane angles, and having a common vertex, be such that one lies wholly within the other, the sum of the plane angles bounding the latter will be greater than the sum of the plane angles bounding the former.

21. Given the three plane angles which contain a solid angle. Find by a plane construction, the angle between any two of the containing planes.

22. Two of the three plane angles which form a solid angle, and also the inclination of their planes being given, to find the third plane angle.

23. Three lines not in the same plane meet in a point; if a plane cut these lines at equal distances from the point of intersection, shew that the perpendicular from that point on the plane will meet it in the center of the circle inscribed in the triangle, formed by the portion of the plane intercepted by the planes passing through the lines.

24. If two straight lines be cut by four parallel planes, the two segments intercepted by the first and second planes, have the same ratio to each other as the two segments intercepted by the third and fourth planes.

III.

25. If planes be drawn through the diagonal and two adjacent edges of a cube, they will be inclined to each other at an angle equal to two-thirds of a right angle.

26. A cube is cut by a plane perpendicular to a diagonal plane, and making a given angle with one of the faces of the cube. Find the angle which it makes with the other faces of the cube.

27. Shew that a cube may be cut by a plane, so that the section shall be a square greater in area than the face of the cube in the proportion of 9 to 8.

. 28. Shew that if a cube be raised on one of its angles so that the diagonal passing through that angle shall be perpendicular to the plane which it touches, its projection on that plane will be a regular hexagon.

29. If any point be taken within a given cube, the square described on its distance from the summit of any of the solid angles of the cube, is equal to the sum of the squares described on its several perpendicular distances from the three sides containing that angle.

30. A rectangular parallelopiped is bisected by all the planes drawn through the axis of it.