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And since the rectangle ABEC contains 5 x 4 square units, (Book II, note, p. 100) and that for every linear unit in AD there is a layer of 5 x 4 cubic units corresponding to it;

consequently, there are 5 x 4 x 3 cubic units in the whole parallelopiped AG.

That is, the product of the three numbers which express the number of linear units in the three edges, will give the number of cubic units in the parallelopiped, and therefore will be the arithmetical representation of its volume.

And generally, if AB, AC, AD; instead of 5, 4 and 3, consisted of a, b, and e linear units, it may be shewn, in a similar manner, that the volume of the parallelopiped would contain abe cubic units, and the product abc would be a proper representation of the volume of the parallelopiped.

If the three sides of the figure were equal to one another, or b and c each equal to a, the figure would become a cube, and its volume would be represented by aaa, or a3.

It may easily be shewn algebraically that the volumes of similar rectangular parallelopipeds are proportional to the cubes of their homologous edges.

Let the adjacent edges of two similar parallelopipeds contain a, b, c, and a', b', d, units respectively. Also let V, V', denote their volumes.

Then V= abc, and V' = a'b'c.

But since the parallelopipeds are similar, therefore =

b

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In a similar manner, it may be shewn that the volumes of all similar solid figures bounded by planes, are proportional to the cubes of their homologous edges.

Prop. vI. From the diagram, the following important construction may be made. If from B a perpendicular BF be drawn to the opposite side DE of the triangle DBE, and AF be joined; then AF shall be perpendicular to DE, and the angle AFB measures the inclination of the planes AED and BED.

Prop. xix. It is also obvious, that if three planes intersect one another; and if the first be perpendicular to the second, and the second be perpendicular to the third; the first shall be perpendicular to the third; also the intersections of every two shall be perpendicular to one another. 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 parallelo

* piped, prove that the superficies is equal to

2{(a+b) v + a*b*}
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.

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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 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 twa opposite faces HF, CE.

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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.'

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