31. In an oblique parallelopiped the sum of the squares on the four diagonals, equals the sum of the squares on the twelve edges.

IV.

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

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

34. 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.

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

36. 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.

37. 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 O 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.

V.

38. 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.

39. 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.

40. 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.

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

42. 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.

BOOK XII.

LEMMA I.

If from the greater of two unequal magnitudes, there be taken more than its half, and from the remainder more than its half; and so on: there shall at length remain a magnitude less than the least of the proposed magnitudes. (Book x. Prop. 1.)

Let AB and C be two unequal magnitudes, of which AB is the greater.
If from AB there be taken more than its half,

and from the remainder more than its half, and so on;
there shall at length remain a magnitude less than C.

For

may be multiplied so as at length to become greater than AB. Let it be so multiplied, and let DE its multiple be greater than AB, and let DE be divided into DF, FG, GE, each equal to C. From AB take BH greater than its half, and from the remainder AH take HK greater than its half, and

so on,

until there be as many divisions in AB as there are in DE:
and let the divisions in AB be AK, KH, HB;

and the divisions in DE be DF, FG, GE.
And because DE is greater than AB,

and that EG taken from DE is not greater than its half, but BH taken from AB is greater than its half; therefore the remainder GD is greater than the remainder HA. Again, because GD is greater than HA,

and that GF is not greater than the half of GD,
but HK is greater than the half of HA;

therefore the remainder FD is greater than the remainder AK: and FD is equal to C,

therefore Cis greater than AK; that is, AK is less than C. Q.E.D. And if only the halves be taken away, the same thing may in the same way be demonstrated.

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 inclinations, 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 ἐστὶν ἡ ὑπὸ πλειόνων ἢ δύο γωνιῶν ἐπιπέδων περιεχομένη, μὴ οὐσῶν ἐν τῷ αὐτῷ ἐπιπέδῳ πρὸς ἑνὶ σημείῳ συνισταμένων. The rendering of this definition by Simson may be slightly amended. The word TEPLEXOμivn is rather comprehended or contained than made: and ovvioraμivwv 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 II., 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, BD, 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 II, note, p. 100) and that for every linear unit in AD there is a layer of 5 × 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 Vabc, and V': a'b'c'.

But since the parallelopipeds are similar, therefore

a b с

a'

=

b'

α α' a' a's

=

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

QUESTIONS ON BOOK XI.

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.