Elements, Book 11

By Euclid

Edition: 0.0.0-dev | March 03, 2014

Authority: SCTA

License Availablity: free, Published under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License

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XX

BOOK XI.

DEFINITIONS.

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

3

3 A straight line is at right angles to a plane, when it makes right angles with all the straight lines which meet it and are in the plane.

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4 A plane is at right angles to a plane when the straight lines drawn, in one of the planes, at right angles to the common section of the planes are at right angles to the remaining plane.

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5 The inclination of a straight line to a plane is, assuming a perpendicular drawn from the extremity of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the extremity of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up.

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6 The inclination of a plane to a plane is the acute angle contained by the straight lines drawn at right angles to the common section at the same point, one in each of the planes.

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7 A plane is said to be similarly inclined to a plane as another is to another when the said angles of the inclinations are equal to one another.

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9 Similar solid figures are those contained by similar planes equal in multitude.

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10 Equal and similar solid figures are those contained by similar planes equal in multitude and in magnitude.

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11 A solid angle is the inclination constituted by more than two lines which meet one another and are not in the same surface, towards all the lines.

12 Otherwise: A solid angle is that which is contained by more than two plane angles which are not in the same plane and are constructed to one point.

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13 A pyramid is a solid figure, contained by planes, which is constructed from one plane to one point.

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14 A prism is a solid figure contained by planes two of which, namely those which are opposite, are equal, similar and parallel, while the rest are parallelograms.

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15 When, the diameter of a semicircle remaining fixed, the semicircle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a sphere.

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16 The axis of the sphere is the straight line which remains fixed and about which the semicircle is turned.

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17 The centre of the sphere is the same as that of the semicircle.

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18 A diameter of the sphere is any straight line drawn through the centre and terminated in both directions by the surface of the sphere.

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19 When, one side of those about the right angle in a right-angled triangle remaining fixed, the triangle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cone.

20 And, if the straight line which remains fixed be equal to the remaining side about the right angle which is carried round, the cone will be right-angled ; if less, obtuse-angled ; and if greater, acute-angled.

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21 The axis of the cone is the straight line which remains fixed and about which the triangle is turned.

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22 And the base is the circle described by the straight line which is carried round.

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23 When, one side of those about the right angle in a rectangular parallelogram remaining fixed, the parallelogram is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cylinder.

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24 The axis of the cylinder is the straight line which remains fixed and about which the parallelogram is turned.

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25 And the bases are the circles described by the two sides opposite to one another which are carried round.

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26 Similar cones and cylinders are those in which the axes and the diameters of the bases are proportional.

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

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28 An octahedron is a solid figure contained by eight equal and equilateral triangles.

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29 An icosahedron is a solid figure contained by twenty equal and equilateral triangles.

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30 A dodecahedron is a solid figure contained by twelve equal, equilateral, and equiangular pentagons.

BOOK XI. PROPOSITIONS.

PROPOSITION 1.

31 A part of a straight line cannot be in the plane of reference and a part in a plane more elevated.

32 For, if possible, let a part AB of the straight line ABC be in the plane of reference, and a part BC in a plane more elevated.

33 There will then be in the plane of reference some straight line continuous with AB in a straight line.

34 Let it be BD ; therefore AB is a common segment of the two straight lines ABC , ABD : which is impossible, inasmuch as, if we describe a circle with centre B and distance AB , the diameters will cut off unequal circumferences of the circle.

35 Therefore a part of a straight line cannot be in the plane of reference, and a part in a plane more elevated. Q. E. D.

PROPOSITION 2.

36 If two straight lines cut one another , they are in one plane , and every triangle is in one plane.

37 For let the two straight lines AB , CD cut one another at the point E ; I say that AB , CD are in one plane, and every triangle is in one plane.

38 For let points F , G be taken at random on EC , EB , let CB , FG be joined, and let FH , GK be drawn across; I say first that the triangle ECB is in one plane.

39 For, if part of the triangle ECB , either FHC or GBK , is in the plane of reference, and the rest in another, a part also of one of the straight lines EC , EB will be in the plane of reference, and a part in another.

40 But, if the part FCBG of the triangle ECB be in the plane of reference, and the rest in another, a part also of both the straight lines EC , EB will be in the plane of reference and a part in another: which was proved absurd. [ XI. 1 ]

42 But, in whatever plane the triangle ECB is, in that plane also is each of the straight lines EC , EB , and, in whatever plane each of the straight lines EC , EB is, in that plane are AB , CD also. [ XI. 1 ]

43 Therefore the straight lines AB , CD are in one plane, and every triangle is in one plane. Q. E. D.

PROPOSITION 3.

44 If two planes cut one another , their common section is a straight line.

45 For let the two planes AB , BC cut one another, and let the line DB be their common section; I say that the line DB is a straight line.

46 For, if not, from D to B let the straight line DEB be joined in the plane AB , and in the plane BC the straight line DFB .

47 Then the two straight lines DEB , DFB will have the same extremities, and will clearly enclose an area: which is absurd.

49 Similarly we can prove that neither will there be any other straight line joined from D to B except DB the common section of the planes AB , BC .

PROPOSITION 4.

51 If a straight line be set up at right angles to two straight lines which cut one another , at their common point of section , it will also be at right angles to the plane through them.

52 For let a straight line EF be set up at right angles to the two straight lines AB , CD , which cut one another at the point E , from E ; I say that EF is also at right angles to the plane through AB , CD .

53 For let AE , EB , CE , ED be cut off equal to one another, and let any straight line GEH be drawn across through E , at random; let AD , CB be joined, and further let FA , FG , FD , FC , FH , FB be joined from the point F taken at random on EF >.

54 Now, since the two straight lines AE , ED are equal to the two straight lines CE , EB , and contain equal angles, [ I. 15 ] therefore the base AD is equal to the base CB , and the triangle AED will be equal to the triangle CEB ; [ I. 4 ] so that the angle DAE is also equal to the angle EBC .

55 But the angle AEG is also equal to the angle BEH ; [ I. 15 ] therefore AGE , BEH are two triangles which have two angles equal to two angles respectively, and one side equal to one side, namely that adjacent to the equal angles, that is to say, AE to EB ; therefore they will also have the remaining sides equal to the remaining sides. [ I. 26 ]

57 And, since AE is equal to EB , while FE is common and at right angles, therefore the base FA is equal to the base FB . [ I. 4 ]

59 And, since AD is equal to CB , and FA is also equal to FB , the two sides FA , AD are equal to the two sides FB , BC respectively; and the base FD was proved equal to the base FC ; therefore the angle FAD is also equal to the angle FBC . [ I. 8 ]

60 And since, again, AG was proved equal to BH , and further FA also equal to FB , the two sides FA , AG are equal to the two sides FB , BH .

61 And the angle FAG was proved equal to the angle FBH ; therefore the base FG is equal to the base FH . [ I. 4 ]

62 Now since, again, GE was proved equal to EH , and EF is common, the two sides GE , EF are equal to the two sides HE , EF ; and the base FG is equal to the base FH ; therefore the angle GEF is equal to the angle HEF . [ I. 8 ]

64 Therefore FE is at right angles to GH drawn at random through E .

65 Similarly we can prove that FE will also make right angles with all the straight lines which meet it and are in the plane of reference.

66 But a straight line is at right angles to a plane when it makes right angles with all the straight lines which meet it and are in that same plane; [ XI. Def. 3 ] therefore FE is at right angles to the plane of reference.

67 But the plane of reference is the plane through the straight lines AB , CD .

68 Therefore FE is at right angles to the plane through AB , CD .

PROPOSITION 5.

70 If a straight line be set up at right angles to three straight lines which meet one another , at their common point of section , the three straight lines are in one plane.

71 For let a straight line AB be set up at right angles to the three straight lines BC , BD , BE , at their point of meeting at B ; I say that BC , BD , BE are in one plane.

72 For suppose they are not, but, if possible, let BD , BE be in the plane of reference and BC in one more elevated; let the plane through AB , BC be produced; it will thus make, as common section in the plane of reference, a straight line. [ XI. 3 ]

74 Therefore the three straight lines AB , BC , BF are in one plane, namely that drawn through AB , BC .

75 Now, since AB is at right angles to each of the straight lines BD , BE , therefore AB is also at right angles to the plane through BD , BE . [ XI. 4 ]

76 But the plane through BD , BE is the plane of reference; therefore AB is at right angles to the plane of reference.

77 Thus AB will also make right angles with all the straight lines which meet it and are in the plane of reference. [ XI. Def. 3 ]

78 But BF which is in the plane of reference meets it; therefore the angle ABF is right.

79 But, by hypothesis, the angle ABC is also right; therefore the angle ABF is equal to the angle ABC .

81 Therefore the straight line BC is not in a more elevated plane; therefore the three straight lines BC , BD , BE are in one plane.

82 Therefore, if a straight line be set up at right angles to three straight lines, at their point of meeting, the three straight lines are in one plane. Q. E. D.

PROPOSITION 6.

83 If two straight lines be at right angles to the same plane , the straight lines will be parallel.

84 For let the two straight lines AB , CD be at right angles to the plane of reference; I say that AB is parallel to CD .

85 For let them meet the plane of reference at the points B , D , let the straight line BD be joined, let DE be drawn, in the plane of reference, at right angles to BD , let DE be made equal to AB , and let BE , AE , AD be joined.

86 Now, since AB is at right angles to the plane of reference, it will also make right angles with all the straight lines which meet it and are in the plane of reference. [ XI. Def. 3 ]

87 But each of the straight lines BD , BE is in the plane of reference and meets AB ; therefore each of the angles ABD , ABE is right.

88 For the same reason each of the angles CDB , CDE is also right.

89 And, since AB is equal to DE , and BD is common, the two sides AB , BD are equal to the two sides ED , DB ; and they include right angles; therefore the base AD is equal to the base BE . [ I. 4 ]

90 And, since AB is equal to DE , while AD is also equal to BE , the two sides AB , BE are equal to the two sides ED , DA ; and AE is their common base; therefore the angle ABE is equal to the angle EDA . [ I. 8 ]

91 But the angle ABE is right; therefore the angle EDA is also right; therefore ED is at right angles to DA .

92 But it is also at right angles to each of the straight lines BD , DC ; therefore ED is set up at right angles to the three straight lines BD , DA , DC at their point of meeting; therefore the three straight lines BD , DA , DC are in one plane. [ XI. 5 ]

93 But, in whatever plane DB , DA are, in that plane is AB also, for every triangle is in one plane; [ XI. 2 ] therefore the straight lines AB , BD , DC are in one plane.

94 And each of the angles ABD , BDC is right; therefore AB is parallel to CD . [ I. 28 ]

PROPOSITION 7.

96 If two straight lines be parallel and points be taken at random on each of them , the straight line joining the points is in the same plane with the parallel straight lines.

97 Let AB , CD be two parallel straight lines, and let points E , F be taken at random on them respectively; I say that the straight line joining the points E , F is in the same plane with the parallel straight lines.

98 For suppose it is not, but, if possible, let it be in a more elevated plane as EGF , and let a plane be drawn through EGF ; it will then make, as section in the plane of reference, a straight line. [ XI. 3 ]

99 Let it make it, as EF ; therefore the two straight lines EGF , EF will enclose an area: which is impossible.

100 Therefore the straight line joined from E to F is not in a plane more elevated; therefore the straight line joined from E to F is in the plane through the parallel straight lines AB , CD .

PROPOSITION 8.

102 If two straight lines be parallel , and one of them be at right angles to any plane , the remaining one will also be at right angles to the same plane.

103 Let AB , CD be two parallel straight lines, and let one of them, AB , be at right angles to the plane of reference; I say that the remaining one, CD , will also be at right angles to the same plane.

104 For let AB , CD meet the plane of reference at the points B , D , and let BD be joined; therefore AB , CD , BD are in one plane. [ XI. 7 ]

105 Let DE be drawn, in the plane of reference, at right angles to BD , let DE be made equal to AB , and let BE , AE , AD be joined.

106 Now, since AB is at right angles to the plane of reference, therefore AB is also at right angles to all the straight lines which meet it and are in the plane of reference; [ XI. Def. 3 ] therefore each of the angles ABD , ABE is right.

107 And, since the straight line BD has fallen on the parallels AB , CD , therefore the angles ABD , CDB are equal to two right angles. [ I. 29 ]

108 But the angle ABD is right; therefore the angle CDB is also right; therefore CD is at right angles to BD .

109 And, since AB is equal to DE , and BD is common, the two sides AB , BD are equal to the two sides ED , DB ; and the angle ABD is equal to the angle EDB , for each is right; therefore the base AD is equal to the base BE .

110 And, since AB is equal to DE , and BE to AD , the two sides AB , BE are equal to the two sides ED , DA respectively, and AE is their common base; therefore the angle ABE is equal to the angle EDA .

111 But the angle ABE is right; therefore the angle EDA is also right; therefore ED is at right angles to AD .

112 But it is also at right angles to DB ; therefore ED is also at right angles to the plane through BD , DA . [ XI. 4 ]

113 Therefore ED will also make right angles with all the straight lines which meet it and are in the plane through BD , DA .

114 But DC is in the plane through BD , DA , inasmuch as AB , BD are in the plane through BD , DA , [ XI. 2 ] and DC is also in the plane in which AB , BD are.

115 Therefore ED is at right angles to DC , so that CD is also at right angles to DE .

117 Therefore CD is set up at right angles to the two straight lines DE , DB which cut one another, from the point of section at D ; so that CD is also at right angles to the plane through DE , DB . [ XI. 4 ]

118 But the plane through DE , DB is the plane of reference; therefore CD is at right angles to the plane of reference.

PROPOSITION 9.

120 Straight lines which are parallel to the same straight line and are not in the same plane with it are also parallel to one another.

121 For let each of the straight lines AB , CD be parallel to EF , not being in the same plane with it; I say that AB is parallel to CD .

122 For let a point G be taken at random on EF , and from it let there be drawn GH , in the plane through EF , AB , at right angles to EF , and GK in the plane through FE , CD again at right angles to EF .

123 Now, since EF is at right angles to each of the straight lines GH , GK , therefore EF is also at right angles to the plane through GH , GK . [ XI. 4 ]

124 And EF is parallel to AB ; therefore AB is also at right angles to the plane through HG , GK . [ XI. 8 ]

125 For the same reason CD is also at right angles to the plane through HG , GK ; therefore each of the straight lines AB , CD is at right angles to the plane through HG , GK .

126 But if two straight lines be at right angles to the same plane, the straight lines are parallel; [ XI. 6 ] therefore AB is parallel to CD .

PROPOSITION 10.

127 If two straight lines meeting one another be parallel to two straight lines meeting one another not in the same plane , they will contain equal angles.

128 For let the two straight lines AB , BC meeting one another be parallel to the two straight lines DE , EF meeting one another, not in the same plane; I say that the angle ABC is equal to the angle DEF .

129 For let BA , BC , ED , EF be cut off equal to one another, and let AD , CF , BE , AC , DF be joined.

130 Now, since BA is equal and parallel to ED , therefore AD is also equal and parallel to BE . [ I. 33 ]

131 For the same reason CF is also equal and parallel to BE .

132 Therefore each of the straight lines AD , CF is equal and parallel to BE .

133 But straight lines which are parallel to the same straight line and are not in the same plane with it are parallel to one another; [ XI. 9 ] therefore AD is parallel and equal to CF .

134 And AC , DF join them; therefore AC is also equal and parallel to DF . [ I. 33 ]

135 Now, since the two sides AB , BC are equal to the two sides DE , EF , and the base AC is equal to the base DF , therefore the angle ABC is equal to the angle DEF . [ I. 8 ]

PROPOSITION 11.

137 From a given elevated point to draw a straight line perpendicular to a given plane.

138 Let A be the given elevated point, and the plane of reference the given plane; thus it is required to draw from the point A a straight line perpendicular to the plane of reference.

139 Let any straight line BC be drawn, at random, in the plane of reference, and let AD be drawn from the point A perpendicular to BC . [ I. 12 ]

140 If then AD is also perpendicular to the plane of reference, that which was enjoined will have been done.

141 But, if not, let DE be drawn from the point D at right angles to BC and in the plane of reference, [ I. 11 ] let AF be drawn from A perpendicular to DE , [ I. 12 ] and let GH be drawn through the point F parallel to BC . [ I. 31 ]

142 Now, since BC is at right angles to each of the straight lines DA , DE , therefore BC is also at right angles to the plane through ED , DA . [ XI. 4 ]

143 And GH is parallel to it; but, if two straight lines be parallel, and one of them be at right angles to any plane, the remaining one will also be at right angles to the same plane; [ XI. 8 ] therefore GH is also at right angles to the plane through ED , DA .

144 Therefore GH is also at right angles to all the straight lines which meet it and are in the plane through ED , DA . [ XI. Def. 3 ]

145 But AF meets it and is in the plane through ED , DA ; therefore GH is at right angles to FA , so that FA is also at right angles to GH .

146 But AF is also at right angles to DE ; therefore AF is at right angles to each of the straight lines GH , DE .

147 But, if a straight line be set up at right angles to two straight lines which cut one another, at the point of section, it will also be at right angles to the plane through them; [ XI. 4 ] therefore FA is at right angles to the plane through ED , GH .

148 But the plane through ED , GH is the plane of reference; therefore AF is at right angles to the plane of reference.

149 Therefore from the given elevated point A the straight line AF has been drawn perpendicular to the plane of reference. Q. E. F.

PROPOSITION 12.

150 To set up a straight line at right angles to a given plane from a given point in it.

151 Let the plane of reference be the given plane, and A the point in it; thus it is required to set up from the point A a straight line at right angles to the plane of reference.

152 Let any elevated point B be conceived, from B let BC be drawn perpendicular to the plane of reference, [ XI. 11 ] and through the point A let AD be drawn parallel to BC . [ I. 31 ]

153 Then, since AD , CB are two parallel straight lines, while one of them, BC , is at right angles to the plane of reference, therefore the remaining one, AD , is also at right angles to the plane of reference. [ XI. 8 ]

154 Therefore AD has been set up at right angles to the given plane from the point A in it.

PROPOSITION 13.

155 From the same point two straight lines cannot be set up at right angles to the same plane on the same side.

156 For, if possible, from the same point A let the two straight lines AB , AC be set up at right angles to the plane of reference and on the same side, and let a plane be drawn through BA , AC ; it will then make, as section through A in the plane of reference, a straight line. [ XI. 3 ]

157 Let it make DAE ; therefore the straight lines AB , AC , DAE are in one plane.

158 And, since CA is at right angles to the plane of reference, it will also make right angles with all the straight lines which meet it and are in the plane of reference. [ XI. Def. 3 ]

159 But DAE meets it and is in the plane of reference; therefore the angle CAE is right.

160 For the same reason the angle BAE is also right; therefore the angle CAE is equal to the angle BAE .

PROPOSITION 14.

163 Planes to which the same straight line is at right angles will be parallel.

164 For let any straight line AB be at right angles to each of the planes CD , EF ; I say that the planes are parallel.

166 Let them meet; they will then make, as common section, a straight line. [ XI. 3 ]

167 Let them make GH ; let a point K be taken at random on GH , and let AK , BK be joined.

168 Now, since AB is at right angles to the plane EF , therefore AB is also at right angles to BK which is a straight line in the plane EF produced; [ XI. Def. 3 ] therefore the angle ABK is right.

170 Thus, in the triangle ABK , the two angles ABK , BAK are equal to two right angles: which is impossible. [ I. 17 ]

171 Therefore the planes CD , EF will not meet when produced; therefore the planes CD , EF are parallel. [ XI. Def. 8 ]

172 Therefore planes to which the same straight line is at right angles are parallel. Q. E. D.

PROPOSITION 15.

173 If two straight lines meeting one another be parallel to two straight lines meeting one another , not being in the same plane , the planes through them are parallel.

174 For let the two straight lines AB , BC meeting one another be parallel to the two straight lines DE , EF meeting one another, not being in the same plane; I say that the planes produced through AB , BC and DE , EF will not meet one another.

175 For let BG be drawn from the point B perpendicular to the plane through DE , EF [ XI. 11 ], and let it meet the plane at the point G ; through G let GH be drawn parallel to ED , and GK parallel to EF . [ I. 31 ]

176 Now, since BG is at right angles to the plane through DE , EF , therefore it will also make right angles with all the straight lines which meet it and are in the plane through DE , EF . [ XI. Def. 3 ]

177 But each of the straight lines GH , GK meets it and is in the plane through DE , EF ; therefore each of the angles BGH , BGK is right.

178 And, since BA is parallel to GH , [ XI. 9 ] therefore the angles GBA , BGH are equal to two right angles. [ I. 29 ]

179 But the angle BGH is right; therefore the angle GBA is also right; therefore GB is at right angles to BA .

180 For the same reason GB is also at right angles to BC .

181 Since then the straight line GB is set up at right angles to the two straight lines BA , BC which cut one another, therefore GB is also at right angles to the plane through BA , BC . [ XI. 4 ]

182 But planes to which the same straight line is at right angles are parallel; [ XI. 14 ] therefore the plane through AB , BC is parallel to the plane through DE , EF .

183 Therefore, if two straight lines meeting one another be parallel to two straight lines meeting one another, not in the same plane, the planes through them are parallel. Q. E. D.

PROPOSITION 16.

184 If two parallel planes be cut by any plane , their common sections are parallel.

185 For let the two parallel planes AB , CD be cut by the plane EFGH , and let EF , GH be their common sections; I say that EF is parallel to GH .

186 For, if not, EF , GH will, when produced, meet either in the direction of F , H or of E , G .

187 Let them be produced, as in the direction of F , H , and let them, first, meet at K .

188 Now, since EFK is in the plane AB , therefore all the points on EFK are also in the plane AB . [ XI. 1 ]

189 But K is one of the points on the straight line EFK ; therefore K is in the plane AB .

190 For the same reason K is also in the plane CD ; therefore the planes AB , CD will meet when produced.

191 But they do not meet, because they are, by hypothesis, parallel; therefore the straight lines EF , GH will not meet when produced in the direction of F , H .

192 Similarly we can prove that neither will the straight lines EF , GH meet when produced in the direction of E , G .

193 But straight lines which do not meet in either direction are parallel. [ I. Def. 23 ]

PROPOSITION 17.

196 If two straight lines be cut by parallel planes , they will be cut in the same ratios.

197 For let the two straight lines AB , CD be cut by the parallel planes GH , KL , MN at the points A , E , B and C , F , D ; I say that, as the straight line AE is to EB , so is CF to FD .

198 For let AC , BD , AD be joined, let AD meet the plane KL at the point O , and let EO , OF be joined.

199 Now, since the two parallel planes KL , MN are cut by the plane EBDO , their common sections EO , BD are parallel. [ XI. 16 ]

200 For the same reason, since the two parallel planes GH , KL are cut by the plane AOFC , their common sections AC , OF are parallel. [ id .]

201 And, since the straight line EO has been drawn parallel to BD , one of the sides of the triangle ABD , therefore, proportionally, as AE is to EB , so is AO to OD . [ VI. 2 ]

202 Again, since the straight line OF has been drawn parallel to AC , one of the sides of the triangle ADC , proportionally, as AO is to OD , so is CF to FD . [ id .]

203 But it was also proved that, as AO is to OD , so is AE to EB ; therefore also, as AE is to EB , so is CF to FD . [ V. 11 ]

PROPOSITION 18.

205 If a straight line be at right angles to any plane , all the planes through it will also be at right angles to the same plane.

206 For let any straight line AB be at right angles to the plane of reference; I say that all the planes through AB are also at right angles to the plane of reference.

207 For let the plane DE be drawn through AB , let CE be the common section of the plane DE and the plane of reference, let a point F be taken at random on CE , and from F let FG be drawn in the plane DE at right angles to CE . [ I. 11 ]

208 Now, since AB is at right angles to the plane of reference, AB is also at right angles to all the straight lines which meet it and are in the plane of reference; [ XI. Def. 3 ] so that it is also at right angles to CE ; therefore the angle ABF is right.

209 But the angle GFB is also right; therefore AB is parallel to FG . [ I. 28 ]

210 But AB is at right angles to the plane of reference; therefore FG is also at right angles to the plane of reference. [ XI. 8 ]

211 Now a plane is at right angles to a plane, when the straight lines drawn, in one of the planes, at right angles to the common section of the planes are at right angles to the remaining plane. [ XI. Def. 4 ]

212 And FG , drawn in one of the planes DE at right angles to CE , the common section of the planes, was proved to be at right angles to the plane of reference; therefore the plane DE is at right angles to the plane of reference.

213 Similarly also it can be proved that all the planes through AB are at right angles to the plane of reference.

PROPOSITION 19.

215 If two planes which cut one another be at right angles to any plane , their common section will also be at right angles to the same plane.

216 For let the two planes AB , BC be at right angles to the plane of reference, and let BD be their common section; I say that BD is at right angles to the plane of reference.

217 For suppose it is not, and from the point D let DE be drawn in the plane AB at right angles to the straight line AD , and DF in the plane BC at right angles to CD .

218 Now, since the plane AB is at right angles to the plane of reference, and DE has been drawn in the plane AB at right angles to AD , their common section, therefore DE is at right angles to the plane of reference. [ XI. Def. 4 ]

219 Similarly we can prove that DF is also at right angles to the plane of reference.

220 Therefore from the same point D two straight lines have been set up at right angles to the plane of reference on the same side: which is impossible. [ XI. 13 ]

221 Therefore no straight line except the common section DB of the planes AB , BC can be set up from the point D at right angles to the plane of reference.

PROPOSITION 20.

223 If a solid angle be contained by three plane angles , any two , taken together in any manner , are greater than the remaining one.

224 For let the solid angle at A be contained by the three plane angles BAC , CAD , DAB ; I say that any two of the angles BAC , CAD , DAB , taken together in any manner, are greater than the remaining one.

225 If now the angles BAC , CAD , DAB are equal to one another, it is manifest that any two are greater than the remaining one.

226 But, if not, let BAC be greater, and on the straight line AB , and at the point A on it, let the angle BAE be constructed, in the plane through BA , AC , equal to the angle DAB ; let AE be made equal to AD , and let BEC , drawn across through the point E , cut the straight lines AB , AC at the points B , C ; let DB , DC be joined.

227 Now, since DA is equal to AE , and AB is common, two sides are equal to two sides; and the angle DAB is equal to the angle BAE ; therefore the base DB is equal to the base BE . [ I. 4 ]

228 And, since the two sides BD , DC are greater than BC , [ I. 20 ] and of these DB was proved equal to BE , therefore the remainder DC is greater than the remainder EC .

229 Now, since DA is equal to AE , and AC is common, and the base DC is greater than the base EC , therefore the angle DAC is greater than the angle EAC . [ I. 25 ]

230 But the angle DAB was also proved equal to the angle BAE ; therefore the angles DAB , DAC are greater than the angle BAC .

231 Similarly we can prove that the remaining angles also, taken together two and two, are greater than the remaining one.

PROPOSITION 21.

233 Any solid angle is contained by plane angles less than four right angles.

234 Let the angle at A be a solid angle contained by the plane angles BAC , CAD , DAB ; I say that the angles BAC , CAD , DAB are less than four right angles.

235 For let points B , C , D be taken at random on the straight lines AB , AC , AD respectively, and let BC , CD , DB be joined.

236 Now, since the solid angle at B is contained by the three plane angles CBA , ABD , CBD , any two are greater than the remaining one; [ XI. 20 ] therefore the angles CBA , ABD are greater than the angle CBD .

237 For the same reason the angles BCA , ACD are also greater than the angle BCD , and the angles CDA , ADB are greater than the angle CDB ; therefore the six angles CBA , ABD , BCA , ACD , CDA , ADB are greater than the three angles CBD , BCD , CDB .

238 But the three angles CBD , BDC , BCD are equal to two right angles; [ I. 32 ] therefore the six angles CBA , ABD , BCA , ACD , CDA , ADB are greater than two right angles.

239 And, since the three angles of each of the triangles ABC , ACD , ADB are equal to two right angles, therefore the nine angles of the three triangles, the angles CBA , ACB , BAC , ACD , CDA , CAD , ADB , DBA , BAD are equal to six right angles; and of them the six angles ABC , BCA , ACD , CDA , ADB , DBA are greater than two right angles; therefore the remaining three angles BAC , CAD , DAB containing the solid angle are less than four right angles.

PROPOSITION 22.

241 If there be three plane angles of which two , taken together in any manner , are greater than the remaining one , and they are contained by equal straight lines , it is possible to construct a triangle out of the straight lines joining the extremities of the equal straight lines.

242 Let there be three plane angles ABC , DEF , GHK , of which two, taken together in any manner, are greater than the remaining one, namely the angles ABC , DEF greater than the angle GHK , the angles DEF , GHK greater than the angle ABC , and, further, the angles GHK , ABC greater than the angle DEF ; let the straight lines AB , BC , DE , EF , GH , HK be equal, and let AC , DF , GK be joined; I say that it is possible to construct a triangle out of straight lines equal to AC , DF , GK , that is, that any two of the straight lines AC , DF , GK are greater than the remaining one.

243 Now, if the angles ABC , DEF , GHK are equal to one another, it is manifest that, AC , DF , GK being equal also, it is possible to construct a triangle out of straight lines equal to AC , DF , GK .

244 But, if not, let them be unequal, and on the straight line HK , and at the point H on it, let the angle KHL be constructed equal to the angle ABC ; let HL be made equal to one of the straight lines AB , BC , DE , EF , GH , HK , and let KL , GL be joined.

245 Now, since the two sides AB , BC are equal to the two sides KH , HL , and the angle at B is equal to the angle KHL , therefore the base AC is equal to the base KL . [ I. 4 ]

246 And, since the angles ABC , GHK are greater than the angle DEF , while the angle ABC is equal to the angle KHL , therefore the angle GHL is greater than the angle DEF .

247 And, since the two sides GH , HL are equal to the two sides DE , EF , and the angle GHL is greater than the angle DEF , therefore the base GL is greater than the base DF . [ I. 24 ]

250 But KL is equal to AC ; therefore AC , GK are greater than the remaining straight line DF .

251 Similarly we can prove that AC , DF are greater than GK , and further DF , GK are greater than AC .

252 Therefore it is possible to construct a triangle out of straight lines equal to AC , DF , GK . Q. E. D.

PROPOSITION 23.

253 To construct a solid angle out of three plane angles two of which , taken together in any manner , are greater than the remaining one: thus the three angles must be less than four right angles.

254 Let the angles ABC , DEF , GHK be the three given plane angles, and let two of these, taken together in any manner, be greater than the remaining one, while, further, the three are less than four right angles; thus it is required to construct a solid angle out of angles equal to the angles ABC , DEF , GHK .

255 Let AB , BC , DE , EF , GH , HK be cut off equal to one another, and let AC , DF , GK be joined; it is therefore possible to construct a triangle out of straight lines equal to AC , DF , GK . [ XI. 22 ]

256 Let LMN be so constructed that AC is equal to LM , DF to MN , and further GK to NL , let the circle LMN be described about the triangle LMN , let its centre be taken, and let it be O ; let LO , MO , NO be joined; I say that AB is greater than LO .

259 Then, since AB is equal to LO , while AB is equal to BC , and OL to OM , the two sides AB , BC are equal to the two sides LO , OM respectively; and, by hypothesis, the base AC is equal to the base LM ; therefore the angle ABC is equal to the angle LOM . [ I. 8 ]

260 For the same reason the angle DEF is also equal to the angle MON , and further the angle GHK to the angle NOL ; therefore the three angles ABC , DEF , GHK are equal to the three angles LOM , MON , NOL .

261 But the three angles LOM , MON , NOL are equal to four right angles; therefore the angles ABC , DEF , GHK are equal to four right angles.

262 But they are also, by hypothesis, less than four right angles: which is absurd.

265 For, if possible, let it be so, and let OP be made equal to AB , and OQ equal to BC , and let PQ be joined.

266 Then, since AB is equal to BC , OP is also equal to OQ , so that the remainder LP is equal to QM .

267 Therefore LM is parallel to PQ , [ VI. 2 ] and LMO is equiangular with PQO ; [ I. 29 ] therefore, as OL is to LM , so is OP to PQ ; [ VI. 4 ] and alternately, as LO is to OP , so is LM to PQ . [ V. 16 ]

268 But LO is greater than OP ; therefore LM is also greater than PQ .

269 But LM was made equal to AC ; therefore AC is also greater than PQ .

270 Since, then, the two sides AB , BC are equal to the two sides PO , OQ , and the base AC is greater than the base PQ , therefore the angle ABC is greater than the angle POQ . [ I. 25 ]

271 Similarly we can prove that the angle DEF is also greater than the angle MON , and the angle GHK greater than the angle NOL .

272 Therefore the three angles ABC , DEF , GHK are greater than the three angles LOM , MON , NOL .

273 But, by hypothesis, the angles ABC , DEF , GHK are less than four right angles; therefore the angles LOM , MON , NOL are much less than four right angles.

274 But they are also equal to four right angles: which is absurd.

276 And it was proved that neither is it equal; therefore AB is greater than LO .

277 Let then OR be set up from the point O at right angles to the plane of the circle LMN , [ XI. 12 ] and let the square on OR be equal to that area by which the square on AB is greater than the square on LO ; [Lemma] let RL , RM , RN be joined.

278 Then, since RO is at right angles to the plane of the circle LMN , therefore RO is also at right angles to each of the straight lines LO , MO , NO .

279 And, since LO is equal to OM , while OR is common and at right angles, therefore the base RL is equal to the base RM . [ I. 4 ]

280 For the same reason RN is also equal to each of the straight lines RL , RM ; therefore the three straight lines RL , RM , RN are equal to one another.

281 Next, since by hypothesis the square on OR is equal to that area by which the square on AB is greater than the square on LO , therefore the square on AB is equal to the squares on LO , OR .

282 But the square on LR is equal to the squares on LO , OR , for the angle LOR is right; [ I. 47 ] therefore the square on AB is equal to the square on RL ; therefore AB is equal to RL .

283 But each of the straight lines BC , DE , EF , GH , HK is equal to AB , while each of the straight lines RM , RN is equal to RL ; therefore each of the straight lines AB , BC , DE , EF , GH , HK is equal to each of the straight lines RL , RM , RN .

284 And, since the two sides LR , RM are equal to the two sides AB , BC , and the base LM is by hypothesis equal to the base AC , therefore the angle LRM is equal to the angle ABC . [ I. 8 ]

285 For the same reason the angle MRN is also equal to the angle DEF , and the angle LRN to the angle GHK .

286 Therefore, out of the three plane angles LRM , MRN , LRN , which are equal to the three given angles ABC , DEF , GHK , the solid angle at R has been constructed, which is contained by the angles LRM , MRN , LRN . Q. E. F.

LEMMA.

287 But how it is possible to take the square on OR equal to that area by which the square on AB is greater than the square on LO , we can show as follows.

288 Let the straight lines AB , LO be set out, and let AB be the greater; let the semicircle ABC be described on AB , and into the semicircle ABC let AC be fitted equal to the straight line LO , not being greater than the diameter AB ; [ IV. 1 ] let CB be joined

289 Since then the angle ACB is an angle in the semicircle ACB , therefore the angle ACB is right. [ III. 31 ]

290 Therefore the square on AB is equal to the squares on AC , CB . [ I. 47 ]

291 Hence the square on AB is greater than the square on AC by the square on CB .

293 Therefore the square on AB is greater than the square on LO by the square on CB .

294 If then we cut off OR equal to BC , the square on AB will be greater than the square on LO by the square on OR . Q. E. F.

PROPOSITION 24.

295 If a solid be contained by parallel planes , the opposite planes in it are equal and parallelogrammic.

296 For let the solid CDHG be contained by the parallel planes AC , GF , AH , DF , BF , AE ; I say that the opposite planes in it are equal and parallelogrammic.

297 For, since the two parallel planes BG , CE are cut by the plane AC , their common sections are parallel. [ XI. 16 ]

299 Again, since the two parallel planes BF , AE are cut by the plane AC , their common sections are parallel. [ XI. 16 ]

301 But AB was also proved parallel to DC ; therefore AC is a parallelogram.

302 Similarly we can prove that each of the planes DF , FG , GB , BF , AE is a parallelogram.

304 Then, since AB is parallel to DC , and BH to CF , the two straight lines AB , BH which meet one another are parallel to the two straight lines DC , CF which meet one another, not in the same plane; therefore they will contain equal angles; [ XI. 10 ] therefore the angle ABH is equal to the angle DCF .

305 And, since the two sides AB , BH are equal to the two sides DC , CF , [ I. 34 ] and the angle ABH is equal to the angle DCF , therefore the base AH is equal to the base DF , and the triangle ABH is equal to the triangle DCF . [ I. 4 ]

306 And the parallelogram BG is double of the triangle ABH , and the parallelogram CE double of the triangle DCF ; [ I. 34 ] therefore the parallelogram BG is equal to the parallelogram CE .

307 Similarly we can prove that AC is also equal to GF , and AE to BF .

PROPOSITION 25.

309 If a parallelepipedal solid be cut by a plane which is parallel to the opposite planes , then , as the base is to the base , so will the solid be to the solid.

310 For let the parallelepipedal solid ABCD be cut by the plane FG which is parallel to the opposite planes RA , DH ; I say that, as the base AEFV is to the base EHCF , so is the solid ABFU to the solid EGCD .

311 For let AH be produced in each direction, let any number of straight lines whatever, AK , KL , be made equal to AE , and any number whatever, HM , MN , equal to EH ; and let the parallelograms LP , KV , HW , MS and the solids LQ , KR , DM , MT be completed.

312 Then, since the straight lines LK , KA , AE are equal to one another, the parallelograms LP , KV , AF are also equal to one another, KO , KB , AG are equal to one another, and further LX , KQ , AR are equal to one another, for they are opposite. [ XI. 24 ]

313 For the same reason the parallelograms EC , HW , MS are also equal to one another, HG , HI , IN are equal to one another, and further DH , MY , NT are equal to one another.

314 Therefore in the solids LQ , KR , AU three planes are equal to three planes.

315 But the three planes are equal to the three opposite; therefore the three solids LQ , KR , AU are equal to one another.

316 For the same reason the three solids ED , DM , MT are also equal to one another.

317 Therefore, whatever multiple the base LF is of the base AF , the same multiple also is the solid LU of the solid AU .

318 For the same reason, whatever multiple the base NF is of the base FH , the same multiple also is the solid NU of the solid HU .

319 And, if the base LF is equal to the base NF , the solid LU is also equal to the solid NU ; if the base LF exceeds the base NF , the solid LU also exceeds the solid NU ; and, if one falls short, the other falls short.

320 Therefore, there being four magnitudes, the two bases AF , FH , and the two solids AU , UH , equimultiples have been taken of the base AF and the solid AU , namely the base LF and the solid LU , and equimultiples of the base HF and the solid HU , namely the base NF and the solid NU , and it has been proved that, if the base LF exceeds the base FN , the solid LU also exceeds the solid NU , if the bases are equal, the solids are equal, and if the base falls short, the solid falls short.

321 Therefore, as the base AF is to the base FH , so is the solid AU to the solid UH . [ V. Def. 5 ] Q. E. D.

PROPOSITION 26.

322 On a given straight line, and at a given point on it , to construct a solid angle equal to a given solid angle.

323 Let AB be the given straight line, A the given point on it, and the angle at D , contained by the angles EDC , EDF , FDC , the given solid angle; thus it is required to construct on the straight line AB , and at the point A on it, a solid angle equal to the solid angle at D .

324 For let a point F be taken at random on DF , let FG be drawn from F perpendicular to the plane through ED , DC , and let it meet the plane at G , [ XI. 11 ] let DG be joined, let there be constructed on the straight line AB and at the point A on it the angle BAL equal to the angle EDC , and the angle BAK equal to the angle EDG , [ I. 23 ] let AK be made equal to DG , let KH be set up from the point K at right angles to the plane through BA , AL , [ XI. 12 ] let KH be made equal to GF , and let HA be joined; I say that the solid angle at A , contained by the angles BAL , BAH , HAL is equal to the solid angle at D contained by the angles EDC , EDF , FDC .

325 For let AB , DE be cut off equal to one another, and let HB , KB , FE , GE be joined.

326 Then, since FG is at right angles to the plane of reference, it will also make right angles with all the straight lines which meet it and are in the plane of reference; [ XI. Def. 3 ] therefore each of the angles FGD , FGE is right.

327 For the same reason each of the angles HKA , HKB is also right.

328 And, since the two sides KA , AB are equal to the two sides GD , DE respectively, and they contain equal angles, therefore the base KB is equal to the base GE . [ I. 4 ]

329 But KH is also equal to GF , and they contain right angles; therefore HB is also equal to FE . [ I. 4 ]

330 Again, since the two sides AK , KH are equal to the two sides DG , GF , and they contain right angles, therefore the base AH is equal to the base FD . [ I. 4 ]

331 But AB is also equal to DE ; therefore the two sides HA , AB are equal to the two sides DF , DE .

332 And the base HB is equal to the base FE ; therefore the angle BAH is equal to the angle EDF . [ I. 8 ]

333 For the same reason the angle HAL is also equal to the angle FDC .

335 Therefore on the straight line AB , and at the point A on it, a solid angle has been constructed equal to the given solid angle at D . Q. E. F.

PROPOSITION 27.

336 On a given straight line to describe a parallelepipedal solid similar and similarly situated to a given parallelepipedal solid.

337 Let AB be the given straight line and CD the given parallelepipedal solid; thus it is required to describe on the given straight line AB a parallelepipedal solid similar and similarly situated to the given parallelepipedal solid CD .

338 For on the straight line AB and at the point A on it let the solid angle, contained by the angles BAH , HAK , KAB , be constructed equal to the solid angle at C , so that the angle BAH is equal to the angle ECF , the angle BAK equal to the angle ECG , and the angle KAH to the angle GCF ; and let it be contrived that, as EC is to CG , so is BA to AK , and, as GC is to CF , so is KA to AH . [ VI. 12 ]

339 Therefore also, ex aequali , as EC is to CF , so is BA to AH . [ V. 22 ]

340 Let the parallelogram HB and the solid AL be completed.

341 Now since, as EC is to CG , so is BA to AK , and the sides about the equal angles ECG , BAK are thus proportional, therefore the parallelogram GE is similar to the parallelogram KB .

342 For the same reason the parallelogram KH is also similar to the parallelogram GF , and further FE to HB ; therefore three parallelograms of the solid CD are similar to three parallelograms of the solid AL .

343 But the former three are both equal and similar to the three opposite parallelograms, and the latter three are both equal and similar to the three opposite parallelograms; therefore the whole solid CD is similar to the whole solid AL . [ XI. Def. 9 ]

344 Therefore on the given straight line AB there has been described AL similar and similarly situated to the given parallelepipedal solid CD . Q. E. F.

PROPOSITION 28.

345 If a parallelepipedal solid be cut by a plane through the diagonals of the opposite planes , the solid will be bisected by the plane.

346 For let the parallelepipedal solid AB be cut by the plane CDEF through the diagonals CF , DE of opposite planes; I say that the solid AB will be bisected by the plane CDEF .

347 For, since the triangle CGF is equal to the triangle CFB , [ I. 34 ] and ADE to DEH , while the parallelogram CA is also equal to the parallelogram EB , for they are opposite, and GE to CH , therefore the prism contained by the two triangles CGF , ADE and the three parallelograms GE , AC , CE is also equal to the prism contained by the two triangles CFB , DEH and the three parallelograms CH , BE , CE ; for they are contained by planes equal both in multitude and in magnitude. [ XI. Def. 10 ]

348 Hence the whole solid AB is bisected by the plane CDEF . Q. E. D.

PROPOSITION 29.

349 Parallelepipedal solids which are on the same base and of the same height , and in which the extremities of the sides which stand up are on the same straight lines , are equal to one another.

350 Let CM , CN be parallelepipedal solids on the same base AB and of the same height, and let the extremities of their sides which stand up, namely AG , AF , LM , LN , CD , CE , BH , BK , be on the same straight lines FN , DK ; I say that the solid CM is equal to the solid CN .

351 For, since each of the figures CH , CK is a parallelogram, CB is equal to each of the straight lines DH , EK , [ I. 34 ] hence DH is also equal to EK .

352 Let EH be subtracted from each; therefore the remainder DE is equal to the remainder HK .

353 Hence the triangle DCE is also equal to the triangle HBK , [ I. 8, 4 ] and the parallelogram DG to the parallelogram HN . [ I. 36 ]

354 For the same reason the triangle AFG is also equal to the triangle MLN .

355 But the parallelogram CF is equal to the parallelogram BM , and CG to BN , for they are opposite; therefore the prism contained by the two triangles AFG , DCE and the three parallelograms AD , DG , CG is equal to the prism contained by the two triangles MLN , HBK and the three parallelograms BM , HN , BN .

356 Let there be added to each the solid of which the parallelogram AB is the base and GEHM its opposite; therefore the whole parallelepipedal solid CM is equal to the whole parallelepipedal solid CN .

PROPOSITION 30.

358 Parallelepipedal solids which are on the same base and of the same height , and in which the extremities of the sides which stand up are not on the same straight lines , are equal to one another.

359 Let CM , CN be parallelepipedal solids on the same base AB and of the same height, and let the extremities of their sides which stand up, namely AF , AG , LM , LN , CD , CE , BH , BK , not be on the same straight lines; I say that the solid CM is equal to the solid CN .

360 For let NK , DH be produced and meet one another at R , and further let FM , GE be produced to P , Q ; let AO , LP , CQ , BR be joined.

361 Then the solid CM , of which the parallelogram ACBL is the base, and FDHM its opposite, is equal to the solid CP , of which the parallelogram ACBL is the base, and OQRP its opposite; for they are on the same base ACBL and of the same height, and the extremities of their sides which stand up, namely AF , AO , LM , LP , CD , CQ , BH , BR , are on the same straight lines FP , DR . [ XI. 29 ]

362 But the solid CP , of which the parallelogram ACBL is the base, and OQRP its opposite, is equal to the solid CN , of which the parallelogram ACBL is the base and GEKN its opposite; for they are again on the same base ACBL and of the same height, and the extremities of their sides which stand up, namely AG , AO , CE , CQ , LN , LP , BK , BR , are on the same straight lines GQ , NR .

PROPOSITION 31.

365 Parallelepipedal solids which are on equal bases and of the same height are equal to one another.

366 Let the parallelepipedal solids AE , CF , of the same height, be on equal bases AB , CD .

368 First, let the sides which stand up, HK , BE , AG , LM , PQ , DF , CO , RS , be at right angles to the bases AB , CD ; let the straight line RT be produced in a straight line with CR ; on the straight line RT , and at the point R on it, let the angle TRU be constructed equal to the angle ALB , [ I. 23 ] let RT be made equal to AL , and RU equal to LB , and let the base RW and the solid XU be completed.

369 Now, since the two sides TR , RU are equal to the two sides AL , LB , and they contain equal angles, therefore the parallelogram RW is equal and similar to the parallelogram HL .

370 Since again AL is equal to RT , and LM to RS , and they contain right angles, therefore the parallelogram RX is equal and similar to the parallelogram AM .

371 For the same reason LE is also equal and similar to SU ; therefore three parallelograms of the solid AE are equal and similar to three parallelograms of the solid XU .

372 But the former three are equal and similar to the three opposite, and the latter three to the three opposite; [ XI. 24 ] therefore the whole parallelepipedal solid AE is equal to the whole parallelepipedal solid XU . [ XI. Def. 10 ]

373 Let DR , WU be drawn through and meet one another at Y , let aTb be drawn through T parallel to DY , let PD be produced to a , and let the solids YX , RI be completed.

374 Then the solid XY , of which the parallelogram RX is the base and Yc its opposite, is equal to the solid XU of which the parallelogram RX is the base and UV its opposite, for they are on the same base RX and of the same height, and the extremities of their sides which stand up, namely RY , RU , Tb , TW , Se , Sd , Xc , XV , are on the same straight lines YW , eV . [ XI. 29 ]

375 But the solid XU is equal to AE : therefore the solid XY is also equal to the solid AE .

376 And, since the parallelogram RUWT is equal to the parallelogram YT for they are on the same base RT and in the same parallels RT , YW , [ I. 35 ] while RUWT is equal to CD , since it is also equal to AB , therefore the parallelogram YT is also equal to CD .

377 But DT is another parallelogram; therefore, as the base CD is to DT , so is YT to DT . [ V. 7 ]

378 And, since the parallelepipedal solid CI has been cut by the plane RF which is parallel to opposite planes, as the base CD is to the base DT , so is the solid CF to the solid RI . [ XI. 25 ]

379 For the same reason, since the parallelepipedal solid YI has been cut by the plane RX which is parallel to opposite planes, as the base YT is to the base TD , so is the solid YX to the solid RI . [ XI. 25 ]

380 But, as the base CD is to DT , so is YT to DT ; therefore also, as the solid CF is to the solid RI , so is the solid YX to RI . [ V. 11 ]

381 Therefore each of the solids CF , YX has to RI the same ratio; therefore the solid CF is equal to the solid YX . [ V. 9 ]

382 But YX was proved equal to AE ; therefore AE is also equal to CF .

383 Next, let the sides standing up, AG , HK , BE , LM , CN , PQ , DF , RS , not be at right angles to the bases AB , CD ; I say again that the solid AE is equal to the solid CF .

384 For from the points K , E , G , M , Q , F , N , S let KO , ET , GU , MV , QW , FX , NY , SI be drawn perpendicular to the plane of reference, and let them meet the plane at the points O , T , U , V , W , X , Y , I , and let OT , OU , UV , TV , WX , WY , YI , IX be joined.

385 Then the solid KV is equal to the solid QI , for they are on the equal bases KM , QS and of the same height, and their sides which stand up are at right angles to their bases. [First part of this Prop.]

386 But the solid KV is equal to the solid AE , and QI to CF ; for they are on the same base and of the same height, while the extremities of their sides which stand up are not on the same straight lines. [ XI. 30 ]

387 Therefore the solid AE is also equal to the solid CF .