# Book:Euclid/The Elements/Book XI

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## Euclid:

## Euclid: *The Elements: Book XI*

Published $\text {c. 300 B.C.E}$

### Contents

Book $\text{XI}$: Spatial Geometry

- Proposition $1$: Straight Line cannot be in Two Planes
- Proposition $2$: Two Intersecting Straight Lines are in One Plane
- Proposition $3$: Common Section of Two Planes is Straight Line
- Proposition $4$: Line Perpendicular to Two Intersecting Lines is Perpendicular to their Plane
- Proposition $5$: Three Intersecting Lines Perpendicular to Another Line are in One Plane
- Proposition $6$: Two Lines Perpendicular to Same Plane are Parallel
- Proposition $7$: Line joining Points on Parallel Lines is in Same Plane
- Proposition $8$: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane
- Proposition $9$: Lines Parallel to Same Line not in Same Plane are Parallel to each other
- Proposition $10$: Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles
- Proposition $11$: Construction of Straight Line Perpendicular to Plane from point not on Plane
- Proposition $12$: Construction of Straight Line Perpendicular to Plane from point on Plane
- Proposition $13$: Straight Line Perpendicular to Plane from Point is Unique
- Proposition $14$: Planes Perpendicular to same Straight Line are Parallel
- Proposition $15$: Planes through Parallel Pairs of Meeting Lines are Parallel
- Proposition $16$: Common Sections of Parallel Planes with other Plane are Parallel
- Proposition $17$: Straight Lines cut in Same Ratio by Parallel Planes
- Proposition $18$: Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane
- Proposition $19$: Common Section of Planes Perpendicular to other Plane is Perpendicular to that Plane
- Proposition $20$: Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle
- Proposition $21$: Solid Angle contained by Plane Angles is Less than Four Right Angles
- Proposition $22$: Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle
- Proposition $23$: Construction of Solid Angle from Three Plane Angles any Two of which are Greater than Other Angle
- Proposition $24$: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms
- Proposition $25$: Parallelepiped cut by Plane Parallel to Opposite Planes
- Proposition $26$: Construction of Solid Angle equal to Given Solid Angle
- Proposition $27$: Construction of Parallelepiped Similar to Given Parallelepiped
- Proposition $28$: Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected
- Proposition $29$: Parallelepipeds on Same Base and Same Height whose Extremities are on Same Lines are Equal in Volume
- Proposition $30$: Parallelepipeds on Same Base and Same Height whose Extremities are not on Same Lines are Equal in Volume
- Proposition $31$: Parallelepipeds on Equal Bases and Same Height are Equal in Volume
- Proposition $32$: Parallelepipeds of Same Height have Volume Proportional to Bases
- Proposition $33$: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides
- Proposition $34$: Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights
- Proposition $35$: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles
- Proposition $36$: Parallelepiped formed from Three Proportional Lines equal to Equilateral Parallelepiped with Equal Angles to it formed on Mean
- Proposition $37$: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional
- Proposition $38$: Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube
- Proposition $39$: Prisms of equal Height with Parallelogram and Triangle as Base