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a plane is equal to the inclination of the straight line to its projection on the plane. If, however, the line be parallel to the plane, the projection of the line is of the same length as the line itself; in all other cases the projection of the line is less than the line, being the base of a right-angled triangle, the hypotenuse of which is the line itself.

The inclination of two lines to each other, which do not meet, is measured by the angle contained by two lines drawn through the same point and parallel to the two given lines.

Def. vi. Planes are distinguished from one another by their inclina. tions, and the inclinations of two planes to one another will be found to be measured by the acute angle formed by two straight lines drawn in the planes, and perpendicular to the straight line which is the common intersection of the two planes.

It is also obvious that the inclination of one plane to another will be measured by the angle contained between two straight lines drawn from the same point, and perpendicular, one on each of the two planes.

The intersection of two planes suggests a new conception of the straight line.

Def. IX. Στερεά γωνία εστίν η υπό πλειόνων ή δύο γωνιών επιπέδων περιεχομένη, μή ουσών εν τω αυτώ επιπέδω προς ένα σημείο συνισταμένων. The rendering of this definition by Simson may be slightly amended. The word teple youévn is rather comprehended or contained than made : and OUVLOTAMÉvwv means joined and fitted together, not meeting. “A solid angle is that which is contained by more than two plane angles joined together at one point, (but) which are not in the same plane.”

When a solid angle is contained by three plane angles, each plane which contains one plane angle, is fixed by the position of the other two, and consequently, only one solid angle can be formed by three plane angles. But when a solid angle is formed by more than three plane angles, if one of the planes be considered fixed in position, there are no conditions which fix the position of the rest of the planes which contain the solid angle, and hence, an indefinite number of solid angles, unequal to one another, may be formed by the same plane angles, when the number of plane angles is more than three.

Def. A. Parallelopipeds are solid figures in some respects analogous to parallelograms, and remarks might be made on parallelopipeds similar to those which were made on rectangular parallelograms in the notes to Book 11., p. 99; and every right-angled parallelopiped may be said to be contained by any three of the straight lines which contain the three right angles by which any one of the solid angles of the figure is formed ; or more briefly, by the three adjacent edges of the parallelopiped.

As all lines are measured by lines, and all surfaces by surfaces, so all solids are measured by solids. The cube is the figure assumed as the measure of solids or volumes, and the unit of volume is that cube, the edge of which is one unit in length.

If the edges of a rectangular parallelopiped can be divided into units of the same length, a numerical expression for the number of cubic units in the parallelopiped may be found, by a process similar to that by which a numerical expression for the area of a rectangle was found.

Let AB, AC, AD be the adjacent edges of a rectangular parallelopiped AG, and let AB contain 5 units, AC, 4 units, and AD, 3 units in length.

Then if through the points of division of AB, AC, AD, planes be drawn parallel to the faces BG, BI), AE respectively, the parallelopiped will be divided into cubic units, all equal to one another.

And since the rectangle ABEC contains 5 x 4 square units, (Book it, 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 c linear units, it may be shewn, in a similar manner, that the volume of the parallelopiped would contain abc 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 a a a, 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', c', units respectively. Also let V, V', denote their volumes.

Then V= abc, and V' = a'b'd'.
But since the parallelopipeds are similar, therefore = =;

у abе a b c а а а аз 13 ся
Hence 7: = a'b'd' =ã7ēsããã = 23 = 7'3 = 25°

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 requi. site 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 uf 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 fol. lowed 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 saine 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

2{(a+b) v + a® 62} 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 siinilar?

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.

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 requi. site 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 uf 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 fol. lowed 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 ... two parts together with thrice the right parallelopiped con

* rectangle and the whole line. 23. When a cube

obliquely to any of its sides, the section will be a r

ram, always greater than a side of the cube, if made

2 sides.

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