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.

33. If a solid angle be formed at A by three plane angles BAC, BAD, CAD; the three planes which bisect the three angles contained by the planes ABC, ACD; ACD, ADB; ADB, ABC, respectively, intersect each other in a straight line passing through A.

34. 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.

35. If a polygon having only salient angles lie within another, and these polygons be made the bases of pyramids having a common vertex, the sum of the plane angles at the vertex of the outer pyramid will be greater than the sum of those at the vertex of the inner.

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

37. 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.

VI.

38. If a straight line be divided into two parts, the cube of the whole line is equal to the cubes of the two parts, together with thrice the right parallelopiped contained by their rectangle and the whole

line.

39. 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.

40. When a cube is cut by a plane obliquely to any of its sides, the section will be an oblong, always greater than the side, if made by cutting opposite sides. Draw a plane cutting two adjacent sides so that the section shall be equal and similar to the side.

41. 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.

42. Show 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.

· 43. Show 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.

44. If a four-sided solid be cut off from a given cube, by a plane passing through the three sides which contain any one of its solid angles, the square of the number of standard units in the base of this solid, shall be equal to the sum of the squares of the numbers of similar units contained in each of its sides.

45. 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 perpendi cular distances from the three sides containing that angle.

46. Trisect a cube.

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

48. If three limited straight lines be parallel, and planes pass through each two of them, and the extremities be joined, a prism will be formed, the ends of which will be parallel if the straight lines be equal.

49. Given the lengths and positions of two straight lines which do not meet when produced and are not parallel; form a parallelopiped of which these two lines shall be two of the edges.

50. The content of a rectangular parallelopiped whose length is any multiple of the breadth, and breadth the same multiple of the depth, is the same as that of the cube whose edge is the breadth.

51. If a right-angled triangular prism be cut by a plane, the volume of the truncated part is equal to a prism of the same base and of height equal to one-third of the sum of the three edges.

52. In an oblique parallelopiped the sum of the squares on the four diagonals, equals the sum of the squares on the twelve edges. 53. Construct a rectangular parallelopiped equal to a given cube, and such, that its three edges shall be continued proportionals,

VII.

54. How many triangular pyramids may be formed whose edges are six given straight lines, of which the sum of any three will form a triangle?

55. Having three points given in a plane, find a point above the plane equidistant from them.

56. A, B are two fixed points in space, and CD a constant length of a given straight line; prove that the pyramid formed by joining the four points A, B, C, D is always of the same magnitude, on whatever part of the given line CD be measured.

57. Bisect a triangular pyramid by a plane passing through one of its angles, and cutting one of its sides in a given direction.

.58.

Shew that the six planes passing through one edge of a triangular pyramid and bisecting the opposite edge meet in a point. 59. Shew how to find the content of a pyramid, whatever be the figure of its base, the altitude and area of the base being given.

60. Compare the content of a triangular pyramid with the content of the parallelopiped of whose faces the edges are diagonals.

61. ABC, the base of a pyramid whose vertex is O, is an equilateral triangle, and the angles BOC, COA, AOB are right angles; shew that three times the square on the perpendicular from 0 on ABC, is equal to the square on the perpendicular, from any of the other angular points of the pyramid, on the faces respectively opposite to them.

62. Two triangles have a common base, and their vertices are in a straight line perpendicular to the plane of the one; there are given the vertical angle of the other, the angles made by each of its sides with the plane of the first and the distance of the vertices of the two triangles, to find the common base.

63. ABCDE is a regular pentagon, on AD, AC are described equilateral triangles with a common vertex F; if a plane through BC cut AF, DF, in extreme and mean ratio in G, H, shew that GHCB is a square.

64. If a pyramid with a polygon for its base he cut by a plane parallel to the base, the section will be a polygon similar to the base.

VIII.

65. If a straight line be at right angles to a plane, the intersection of the perpendiculars let fall from the several points of that line on another plane, is a straight line which makes right angles with the common section of the two planes.

66. Find the locus of those points which are equidistant from three given planes.

67. Two planes intersect; shew that the loci of the points, from which perpendiculars on the planes are equal to a given straight line, are straight lines; and that four planes may be drawn, each passing through two of these lines, such that the perpendicular from any point in the line of intersection of the given planes upon any one of the four planes, shall be equal to the given line.

IX.

68. If there be two straight lines which are not parallel, but which do not meet, though produced ever so far both ways, shew that two parallel planes may be determined so as to pass, the one through the one line, the other through the other; and that the perpendicular distance of these planes is the shortest distance of any point that can be taken in the one line from any point taken in the other.

69. Of all the angles, which a straight line makes with any straight lines drawn in a given plane to meet it, the least is that which measures the inclination of the line to the plane.

70. Show that if a straight line meets two others not in the same plane with one another, and is perpendicular to both; the part of it intercepted between them is the shortest line that can be drawn from any point in one of them to any point in the other.

71. Find a point in a given straight line such that the sum of its distances from two given points (not in the same plane with the given straight line) may be the least possible.

72. If, round a line which is drawn from a point in the common section of two planes at right angles to one of them, a third plane be made to revolve, shew that the plane angle made by the three planes is then the greatest, when the revolving plane is perpendicular to each of the two fixed planes.

73. Two points are taken on a wall and joined by a line which passes round a corner of the wall. This line is the shortest when its parts make equal angles with the edge at which the parts of the wall meet.

74. Prove that among all parallelopipeds of given volume, a cube is that which has the least surface.

a

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GEOMETRICAL EXERCISES ON BOOK XII.

THEOREM 1.

If semicircles ADB, BEC be described on the sides AB, BC of a rightangled triangle, and on the hypotenuse another semicircle AFBGC be described, passing through the vertex B; the lunes AFBD and BGCE are together equal to the triangle ABC.

It has been demonstrated (XII. 2.) that the areas of circles are to one another as the squares on their diameters; it follows also that semicircles will be to each other in the same proportion.

Therefore the semicircle ADB is to the semicircle ABC, as the square on AB is to the square on AC,

and the semicircle CEB is to the semicircle ABC as the square on BC is to the square on AC,

hence the semicircles ADB, CEB, are to the semicircle ABC as the squares on AB, BC are to the square on AC;

but the squares on AB, BC are equal to the square on AC: (1. 47.) therefore the semicircles ADB, CEB are equal to the semicircle ABC. (v. 14.)

From these equals take the segments AFB, BGC of the semicircle on AC, and the remainders are equal,

that is, the lunes AFBD, BGCE are equal to the triangle BAC.

THEOREM II.

If on any two segments of the diameter of a semicircle semicircles be described, all towards the same parts, the area included between the_three circumferences (called upßnλos) will be equal to the area of a circle, the diameter of which is a mean proportional between the segments.

Let ABC be a semicircle whose diameter is AB,
and let AB be divided into any two parts in D,

and on AD, DC let two semicircles be described on the same side; also let DB be drawn perpendicular to AC.

Then the area contained between the three semicircles, is equal to the area of the circle whose diameter is BD.

Since AC is divided into two parts in C,

the square on AC is equal to the squares on AD, DC, and twice the rectangle AD, DC; (II. 4.)

and since BD is a mean proportional between AD, DC; the rectangle AD, DC is equal to the square on DB, (vi. 17.) therefore the square on AC is equal to the squares on AD, DC, and twice the square on DB.

DC,

But circles are to one another as the squares on their diameters or radii, (XII. 2.)

therefore the circle whose diameter is AC, is equal to the circles whose diameters are AD, DC, and double the circle whose diameter is BD ;

wherefore the semicircle whose diameter is AC is equal to the circle whose diameter is BD, together with the two semicircles whose diameters are AD and DC:

if the two semicircles whose diameters are AD and DC be taken from these equals,

therefore the figure comprised between the three semi-circumferences is equal to the circle whose diameter is DB.

THEOREM III.

Every section of a sphere by a plane is a circle.

If the plane pass through the center of the sphere, it is manifest that the section is a circle, having the same diameter as the generating semicircle.

But if the cutting plane does not pass through the center,

let AEB be any other section of the sphere made by a plane not passing through the center of the sphere.

Take the center C, and draw the diameter HCK perpendicular to the section AEB, and meeting it in D;

draw AB passing through D, and join AC; take E, F, any other points in the line AEB, and join CE, DE; CF, DF.