Let the lines AB, CD be cut by the parallel planes GH, KL MN, in the points A, E, B, and C, F, D; AE EB:: CF: FD.

Draw AC, BD, AD; let AD meet the plane KL in the point X; draw EX, XF. Because the parallel planes KL, MN are cut by the plane EBDX, the common sections EX, BD are parallel (11. 11); and because the parallel planes GH, KL are cut by the plane AXFC, the common sections AC, XF are parallel.

K

Because the line EX is parallel to BD, a side of the triangle ABD, AE : EB :: AX : XD (2. 6); and because XF is parallel to AC, a side of the triangle ADC, AX: XD :: CF: FD. Therefore AE : M EB: CF: FD (Propor. 34). Wherefore if two &c. Q. E. D.

H

A

L

F

E

X

PROPOSITION XIII. THEOREM.

If a straight line be perpendicular to a plane, every plane which passes through the line is also perpendicular to that plane.

Let the line AB be perp. to a plane CK; every plane which passes through AB is perp. to the plane CK.

D

G A

H

Let any plane DE pass through AB, and let CE be the common section of the planes DE, CK; take any point F in CE, from which draw FG in the plane DE perp. to CE. Because AB is perp. to the plane CK, it is also perp. to every line meeting it in that plane (1 Def.11); consequently it is perp. to CE; wherefore ABF is a right angle. But GFB is a right angle; therefore AB is parallel to FG (Cor. 28. 1). But AB is perp. to the plane CK; therefore FG is also perp. to CK (7. 11).

K

F BE

Now any straight line FG in the plane DE, which is perp. to CE the common section of the planes CK, DE, has been proved to be perp. to the other plane CK; therefore the plane DE is perp. to the plane CK (2 Def. 11). In like manner it may be proved that all the planes which pass through AB are perp. to the plane CK. Therefore, if a straight &c. Q. É. D.

PROPOSITION XIV. THEOREM.

If two planes which cut each other be perpendicular to a third plane, their common section will be perpendicular to the third plane.

Let the two planes AB, BC be perp. to a third plane ADC, and let BD be the common section of AB, BC; BD is perp. to the plane ADC.

From D in the plane ADC draw DE perp. to AD, and DF to DC. Because DE is perp. to AD, the common section of the planes AB and ADC, and because the plane AB is perp. to ADC, DE is perp. to the plane AB (2 Def. 11), and therefore also to the line BD in that plane (1 Def. 11). For the same reason DF is perp. to DB. Since BD is perp. to both the lines DE and DF, it is perp. to the plane in which DE and DF are situate, that is, to the plane ADC (4. 11). Wherefore, if two planes &c. Q. E. D.

A

B

F

E

C

THE PRINCIPAL THEOREMS IN BOOK XI.

Any two straight lines which cut each other are in one plane; and any three straight lines which meet one another in different points are in one plane.

If two planes cut each other, their common section is a straight line.

If a straight line be perpendicular to each of two straight lines in their point of intersection, it will also be perpendicular to the plane in which those lines are situate.

If three straight lines meet in one point, and a straight line be perpendicular to each of them in that point, those three lines are in one and the same plane.

If two straight lines be perpendicular to the same plane they will be parallel to each other.

If two straight lines be parallel, and if one of them be perpendicular to a plane, the other also is perpendicular to the same plane.

Two straight lines which are parallel to the same straight line, and are not both in the same plane with it, are parallel to each other.

If two straight lines which meet each other be parallel to two other straight lines which meet each other toward the same parts, though not in the same plane with the first two, the first two lines and the other two will contain equal angles.

If two parallel planes be cut by a third plane, their common sections with it are parallel.

If two straight lines be cut by parallel planes they will be cut in the same ratio.

If a straight line be perpendicular to a plane, every plane which passes through the line is also perpendicular to that plane.

END OF BOOK XI.

ELEMENTS OF GEOMETRY.

BOOK XII.

OF THE PROPERTIES OF SOLIDS.

DEFINITIONS.

1. A solid is that magnitude which has length, breadth and thickness.

2. Similar solid figures are such as are contained by the same number of similar planes similarly situate, and having like inclinations to one another.

3. A pyramid is a solid figure contained by more than two triangular planes that are constituted between one plane base and a point above it in which the planes meet.

4. A prism is a solid figure contained by plane figures, of which two that are opposite are equal, similar, and parallel to each other; and the rest are parallelograms.

Note.-Prisms and pyramids take particular names according to the figure of their bases. Thus, if the base be a triangle, it is called a triangular prism or pyramid; if a square, it is called a square prism or pyramid.

ED.

5. A parallelopiped is a prism, or solid figure, contained by six quadrilateral figures, whereof every opposite two are pa

rallel.

6. A cube is a solid figure contained by six equal squares.

7. A sphere is a solid figure described by the revolution of a semicircle about a diameter, which remains unmoved: or, a sphere is a solid figure bounded by one curve surface, which is every where equally distant from a certain point within it called the centre.

8. The axis of a sphere is the fixed straight line about which the semicircle revolves.

9. The centre of a sphere is the same with that of a semicircle by which it is described.

10. The diameter of a sphere is any straight line which passes through the centre, and is terminated both ways by the superficies of the sphere.

11. A cone is a solid figure described by the revolution of a straight line round the circumference of a circle, one end of which line is fixed at a point above the plane of the circle.

12. The axis of a cone is the straight line joining the vertex, or fixed point, and the centre of the circle about which the cone is described.

13. The base of a cone is the circle about which the describing line revolves.

14. A cylinder is a solid figure described by the revolution of a rectangle about one of its sides, which remains fixed.

15. The axis of a cylinder is the fixed straight line about which the rectangle revolves.

16. The bases of a cylinder are the circles described by the two revolving opposite sides of the rectangle.

17. Similar cones and cylinders are those which have their axis and the diameters of their bases proportionals.

PROPOSITION I. THEOREM.

If two solids be contained by the same number of equal and similar planes, similarly situated, and if the inclination of any two contiguous planes in one solid be the same with the inclination of the two equal and similarly situated planes in the other, the solids are equal and similar.

Let AG, KQ be two solids contained by the same number of equal and similar planes, similarly situated, so that the plane AC is similar and equal to KM, the plane AF to KP, BG to LQ, GD to QN, DE to NO, and FH to PR; and let the inclination of the plane AF to AC be the same with the inclination of the plane KP to KM, and so of the rest; the solid KQ is equal and similar to the solid AG.