Divide the radius AC into four equal parts; then upon BD, BF, BG, describe semicircumferences. Draw the radius CH at rightangles to the diameter AB; and the points K, L, M, thus determined, will be the points to which they must grind respectively. We have (B. IV, Prop. xvIII, Cor. 1,)

therefore CM2, CL2, CK2, CH2 are to each other as CD, CF, CG, CA, or as 1, 2, 3, 4. Hence the areas of the circles whose radii are CM, CL, CK, CH are to each other as 1, 2, 3, 4, (B. V, Prop. xII;) therefore the stone has in this way been divided equally among four individuals. A similar method would apply for a greater number of divisions.

If we denote the radius of the stone by R, and the number of individuals by n, we shall have

BOOK SIXTH.

DEFINITIONS.

1. THE intersection of two planes is the line in which they meet to cut each other.

2. A line is perpendicular to a plane, when it is perpendicular to all the lines in that plane which meet it.

3. One plane is perpendicular to another, when every line in the one which is perpendicular to their intersection is perpendicular to the other plane.

4. The inclination of two planes to each other, or the angle they form between them, is the angle contained by two lines drawn from any point in their intersection, and at right-angles to the same, one of these lines in each plane.

5. A line is parallel to a plane, when, if both are produced to any distance, they do not meet; and, conversely, the plane is then also parallel to the line.

6. Two planes are parallel to each other, when, both being produced to any distance, they do not meet.

7. A solid angle is the angular space included between three or more planes which meet at the same point.

16

PROPOSITION I.

THEOREM. One part of a straight line. cannot be in a plane, and another part out of it.

For (B. I, Def. VII,) when a straight line has two points common with a plane, it lies wholly in that plane.

PROPOSITION II.

THEOREM. Two straight lines which intersect each other, lie in the same plane, and determine its position.

E

A

B

Let AB, AC be two straight lines which intersect each other in A; and conceive some plane passing through one of the lines as AB, and if also AC be in this plane, then it is clear that the two lines, according to the terms of the proposition, are in the same plane; but if not, let the plane passing through AB be supposed to be turned round AB till it passes through the point C, then the line AC, which has two of its points A and C in this plane, lies wholly in it; and hence the position of the plane is determined by the single condition of containing the two straight lines AB, AC.

Cor. 1. A triangle ABC, or any three points not in a straight line, determines the position of a plane.

Cor. 2. Hence, also, two parallels AB, CD determine the position of a plane; for, drawing the secant EF, the plane of the two straight lines AB, EF is that of the parallels AB, CD.

PROPOSITION III.

THEOREM. The intersection of two planes is a straight line.

Let DC and EF be two planes cutting each other, and A, B two points in which the planes meet. Draw the line AB; this line is the intersection of the two planes.

F

D

B

For, because the straight line touches the two planes in the points A and B, it lies wholly in both these planes, or is common to both of them; that is, the intersection of the two planes is in a straight line

PROPOSITION IV.

THEOREM. If a straight line is perpendicular to each of two straight lines, at their point of intersection, it will be perpendicular to the plane of these lines.