to two right angles [1. XXIX.]; but ABF is right (hyp.); therefore BFG is right, that is, FG is perpendicular to CE. Hence every line in the plane DE, drawn perpendicular to the common section of the planes DE, CI, is normal to the plane CI. Therefore [XI. Def. VIII.] the planes DE, CI are perpendicular to each other.

PROP. XIX.—THEOREM.

If two intersecting planes (AB, BC) be each perpendicular to a third plane (ADC), their common section (BD) shall be normal to that plane.

E

B

Dem.-If not, draw from D in the plane AB the line DE perpendicular to AD, the common section of the planes AB, ADC; and in the plane BC draw BF perpendicular to the common sections DC of the planes BC, ADC. Then because the plane AB is perpendicular to ADC, the line DE in AB is normal to the plane ADC [XI. Def. VIII.

1.]. In like manner DF

is normal to it. Therefore from A

Hence

the point D there are two distinct normals to the plane ADC, which [XI. XIII.] is impossible. BD must be normal to the plane ADC.

Exercises.

1. If three planes have a common line of intersection, the normals drawn to these planes from any point of that line are coplanar.

2. If two intersecting planes be respectively perpendicular to two intersecting lines, the line of intersection of the former is normal to the plane of the latter.

3. In the last case, show that the dihedral angle between the planes is equal to the rectilinear angle between the normals.

PROP. XX.-THEOREM.

The sum of any two plane angles (BAD, DAC) of a trihedral angle (A) is greater than the third (BAC).

D

Dem.-If the third angle BAC be less than or equal to either of the other angles the proposition is evident. If not, suppose it greater: take any point D in AD, and at the point A in the plane BAC make the angle BAE equal BAD [I. xxIII.],

B

E

and cut off AE equal AD. Through E draw BC, cutting AB, AC in the points B, C. Join DB, BC.

Then the triangles BAD, BAE have the two sides BA, AD in one equal respectively to the two sides BA, AE in the other, and the angle BAD equal to BAE; therefore the third side BD is equal to BE. But the sum of the sides BD, DC is greater than AC; hence DC is greater than EC. Again, because the triangles DAC, EAC have the sides DA, AC respectively equal to the sides EA, AC in the other, but the base DC greater than EC; therefore [1. XXV.] the angle DAC is greater than EAC, but the angle DAB is equal to BAE (const.). Hence the sum of the angles BAD, DAC is greater than the angle BAC.

PROP. XXI.-THEOREM.

The sum of all the plane angles (BAC, CAD, &c.) forming any solid angle (A) is less than four right angles.

Dem. Suppose for simplicity that the solid angle A is contained by five plane angles BAC, CAD, DAE, EAF, FAB; and let the planes of these angles be cut

by another plane in the lines BC, CD, DE, EF, FB; then we have [XI. xx.],

many right angles as there are triangles BAC, CAD, &c., that is, equal to ten right angles. Hence the sum of the angles forming the solid angle is less than four right angles.

Observation.-This prop. may not hold if the polygonal base BCDEF contain re-entrant angles.

Exercises on Book XI. .

1. Any face angle of a trihedral angle is less than the sum, but greater than the difference, of the supplements of the other two face angles.

2. A solid angle cannot be formed of equal plane angles which are equal to the angles of a regular polygon of n sides, except in the case of n = 3, 4, or 5.

3. Through one of two non-coplanar lines draw a plane parallel to the other.

4. Draw a common perpendicular to two non-coplanar lines, and show that it is the shortest distance between them.

5. If two of the plane angles of a tetrahedral angle be equal, the planes of these angles are equally inclined to the plane of the third angle, and conversely. If two of the planes of a trahedral angle be equally inclined to the third plane, the angles contained in those planes are equal.

6. The three lines of intersection of three planes are either parallel or concurrent.

7. If a trihedral angle O be formed by three right angles, and A, B, C be points along the edges, the orthocentre of the triangles ABC is the foot of the normal from O on the plane ABC.

8. If through the vertex O of a trihedral angle O-ABC any line OD be drawn interior to the angle, the sum of the rectilineal angles DOA, DOB, DOC is less than the sum, but greater than half the sum, of the face angles of the trihedral.

9. If on the edges of a trihedral angle O-ABC three equal lines OA, OB, OC be taken, each of these is greater than the radius of the circle described about the triangle ABC.

10. Given the three angles of a trihedral angle, find, by a plane construction, the angles between the containing planes.

11. If any plane P cut the four sides of a Gauche quadrilateral (a quadrilateral whose angular points are not coplanar) ABCD in four points, a, b, c, d, then the product of the four ratios

Aa Bb Сс Dd

aB' bC' cD' dA

is plus unity, and conversely, if the product

Aa Bb Cc Dd

aBbCcdA

the points a, b, c, d are coplanar.

+1,

12. If in the last exercise the intersecting plane be parallel to any two sides of the quadrilateral, it cuts the two remaining sides proportionally.

DEF. X.-If at the vertex O of a trihedral angle O-ABC we draw normals OA', OB', OC' to the faces OBC, OCA, OAB, respectively, in such a manner that OA' will be on the same side of the plane OBC as OA, &c., the trihedral angle O-A'B'C' is called the supplementary of the trihedral angle O-ABC.

13. If O-A'B'C' be the supplementary of O-ABC, prove that O-ABC is the supplementary of O—A'B'C'.

14. If two trihedral angles be supplementary, each dihedral angle of one is the supplement of the corresponding face angle of the other.

15. Through a given point draw a right line which will meet two non-coplanar lines.

16. Draw a right line parallel to a given line, which will meet two non-coplanar lines.

17. Being given an angle AOB, the locus of all the points P of space, such that the sum of the projections of the line OP on OA and OB may be constant, is a plane.

APPENDIX.

PRISM, PYRAMID, CYLINDER, SPHERE, AND CONE.

DEFINITIONS.

1. A polyhedron is a solid figure contained by plane figures; if it be contained by four plane figures it is called a tetrahedron; by six, a hexahedron; by eight, an octahedron; by twelve, a dodecahedron; and if by twenty, an icosahedron.

1. If the plane faces of a polyhedron be equal and similar rectilineal figures, it is called a regular polyhedron.

III. A pyramid is a polyhedron of which all the faces but one meet in a point. This point is called the vertex; and the opposite face, the base.

IV. A prism is a polyhedron having a pair of parallel faces which are equal and similar rectilineal figures, and are called its ends. The others, called its side faces, are parallelograms.

v. A prism whose ends are perpendicular to its sides is called a right prism; any other is called an oblique prism.

VI. The altitude of a pyramid is the length of the perpendicular drawn from its vertex to its base; and the altitude of a prism is the perpendicular distance between its ends.

VII. A parallelopiped is a prism whose bases are parallelograms. A parallelopiped is evidently a hexahedron.