Elements, Book 9

By Euclid

Edition: 0.0.0-dev | March 03, 2014

Authority: SCTA

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BOOK IX.

PROPOSITIONS

PROPOSITION 1.

1 If two similar plane numbers by multiplying one another make some number, the product will be square.

2 Let A , B be two similar plane numbers, and let A by multiplying B make C ; I say that C is square.

5 Since then A by multiplying itself has made D , and by multiplying B has made C , therefore, as A is to B , so is D to C . [ VII. 17 ]

6 And, since A , B are similar plane numbers, therefore one mean proportional number falls between A , B . [ VIII. 18 ]

7 But, if numbers fall between two numbers in continued proportion, as many as fall between them, so many also fall between those which have the same ratio; [ VIII. 8 ] so that one mean proportional number falls between D , C also.

8 And D is square; therefore C is also square. [ VIII. 22 ] Q. E. D.

PROPOSITION 2.

9 If two numbers by multiplying one another make a square number, they are similar plane numbers.

10 Let A , B be two numbers, and let A by multiplying B make the square number C ; I say that A , B are similar plane numbers.

11 For let A by multiplying itself make D ; therefore D is square.

12 Now, since A by multiplying itself has made D , and by multiplying B has made C , therefore, as A is to B , so is D to C . [ VII. 17 ]

13 And, since D is square, and C is so also, therefore D , C are similar plane numbers.

14 Therefore one mean proportional number falls between D , C . [ VIII. 18 ]

15 And, as D is to C , so is A to B ; therefore one mean proportional number falls between A , B also. [ VIII. 8 ]

16 But, if one mean proportional number fall between two numbers, they are similar plane numbers; [ VIII. 20 ] therefore A , B are similar plane numbers. Q. E. D.

PROPOSITION 3.

17 If a cube number by multiplying itself make some number, the product will be cube.

18 For let the cube number A by multiplying itself make B ; I say that B is cube.

19 For let C , the side of A , be taken, and let C by multiplying itself make D .

20 It is then manifest that C by multiplying D has made A .

21 Now, since C by multiplying itself has made D , therefore C measures D according to the units in itself.

22 But further the unit also measures C according to the units in it; therefore, as the unit is to C , so is C to D . [ VII. Def. 20 ]

23 Again, since C by multiplying D has made A , therefore D measures A according to the units in C .

24 But the unit also measures C according to the units in it; therefore, as the unit is to C , so is D to A .

25 But, as the unit is to C , so is C to D ; therefore also, as the unit is to C , so is C to D , and D to A .

26 Therefore between the unit and the number A two mean proportional numbers C , D have fallen in continued proportion.

27 Again, since A by multiplying itself has made B , therefore A measures B according to the units in itself.

28 But the unit also measures A according to the units in it; therefore, as the unit is to A , so is A to B . [ VII. Def. 20 ]

29 But between the unit and A two mean proportional numbers have fallen; therefore two mean proportional numbers will also fall between A , B . [ VIII. 8 ]

30 But, if two mean proportional numbers fall between two numbers, and the first be cube, the second will also be cube. [ VIII. 23 ]

PROPOSITION 4.

32 If a cube number by multiplying a cube number make some number, the product will be cube.

33 For let the cube number A by multiplying the cube number B make C ; I say that C is cube.

34 For let A by multiplying itself make D ; therefore D is cube. [ IX. 3 ]

35 And, since A by multiplying itself has made D , and by multiplying B has made C therefore, as A is to B , so is D to C . [ VII. 17 ]

36 And, since A , B are cube numbers, A , B are similar solid numbers.

37 Therefore two mean proportional numbers fall between A , B ; [ VIII. 19 ] so that two mean proportional numbers will fall between D , C also. [ VIII. 8 ]

38 And D is cube; therefore C is also cube [ VIII. 23 ] Q. E. D.

PROPOSITION 5.

39 If a cube number by multiplying any number make a cube number, the multiplied number will also be cube.

40 For let the cube number A by multiplying any number B make the cube number C ; I say that B is cube.

41 For let A by multiplying itself make D ; therefore D is cube. [ IX. 3 ]

42 Now, since A by multiplying itself has made D , and by multiplying B has made C , therefore, as A is to B , so is D to C . [ VII. 17 ]

43 And since D , C are cube, they are similar solid numbers.

44 Therefore two mean proportional numbers fall between D , C . [ VIII. 19 ]

45 And, as D is to C , so is A to B ; therefore two mean proportional numbers fall between A , B also. [ VIII. 8 ]

46 And A is cube; therefore B is also cube. [ VIII. 23 ]

PROPOSITION 6.

47 If a number by multiplying itself make a cube number, it will itself also be cube.

48 For let the number A by multiplying itself make the cube number B ; I say that A is also cube.

50 Since, then, A by multiplying itself has made B , and by multiplying B has made C , therefore C is cube.

51 And, since A by multiplying itself has made B , therefore A measures B according to the units in itself.

52 But the unit also measures A according to the units in it.

53 Therefore, as the unit is to A , so is A to B . [ VII. Def. 20 ]

54 And, since A by multiplying B has made C , therefore B measures C according to the units in A .

55 But the unit also measures A according to the units in it.

56 Therefore, as the unit is to A , so is B to C . [ VII. Def. 20 ]

57 But, as the unit is to A , so is A to B ; therefore also, as A is to B , so is B to C .

58 And, since B , C are cube, they are similar solid numbers.

59 Therefore there are two mean proportional numbers between B , C . [ VIII. 19 ]

61 Therefore there are two mean proportional numbers between A , B also. [ VIII. 8 ]

62 And B is cube; therefore A is also cube. [cf. VIII. 23 ] Q. E. D.

PROPOSITION 7.

63 If a composite number by multiplying any number make some number, the product will be solid.

64 For let the composite number A by multiplying any number B make C ; I say that C is solid.

65 For, since A is composite, it will be measured by some number. [ VII. Def. 13 ]

66 Let it be measured by D ; and, as many times as D measures A , so many units let there be in E .

67 Since then D measures A according to the units in E , therefore E by multiplying D has made A . [ VII. Def. 15 ]

68 And, since A by multiplying B has made C , and A is the product of D , E , therefore the product of D , E by multiplying B has made C .

69 Therefore C is solid, and D , E , B are its sides. Q. E. D.

PROPOSITION 8.

70 If as many numbers as we please beginning from an unit be in continued proportion, the third from the unit will be square, as will also those which successively leave out one; the fourth will be cube, as will also all those which leave out two; and the seventh will be at once cube and square, as will also those which leave out five.

71 Let there be as many numbers as we please, A , B , C , D , E , F , beginning from an unit and in continued proportion; I say that B , the third from the unit, is square, as are also all those which leave out one; C , the fourth, is cube, as are also all those which leave out two; and F , the seventh, is at once cube and square, as are also all those which leave out five.

72 For since, as the unit is to A , so is A to B , therefore the unit measures the number A the same number of times that A measures B . [ VII. Def. 20 ]

73 But the unit measures the number A according to the units in it; therefore A also measures B according to the units in A .

74 Therefore A by multiplying itself has made B ; therefore B is square.

75 And, since B , C , D are in continued proportion, and B is square, therefore D is also square. [ VIII. 22 ]

77 Similarly we can prove that all those which leave out one are square.

78 I say next that C , the fourth from the unit, is cube, as are also all those which leave out two.

79 For since, as the unit is to A , so is B to C , therefore the unit measures the number A the same number of times that B measures C .

80 But the unit measures the number A according to the units in A ; therefore B also measures C according to the units in A .

82 Since then A by multiplying itself has made B , and by multiplying B has made C , therefore C is cube.

83 And, since C , D , E , F are in continued proportion, and C is cube, therefore F is also cube. [ VIII. 23 ]

84 But it was also proved square; therefore the seventh from the unit is both cube and square.

85 Similarly we can prove that all the numbers which leave out five are also both cube and square. Q. E. D.

PROPOSITION 9.

86 If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be square, all the rest will also be square. And, if the number after the unit be cube, all the rest will also be cube.

87 Let there be as many numbers as we please, A , B , C , D , E , F , beginning from an unit and in continued proportion, and let A , the number after the unit, be square; I say that all the rest will also be square.

88 Now it has been proved that B , the third from the unit, is square, as are also all those which leave out one; [ IX. 8 ] I say that all the rest are also square.

89 For, since A , B , C are in continued proportion, and A is square, therefore C is also square. [ VIII. 22 ]

90 Again, since B , C , D are in continued proportion, and B is square, D is also square. [ VIII. 22 ]

91 Similarly we can prove that all the rest are also square.

92 Next, let A be cube; I say that all the rest are also cube.

93 Now it has been proved that C , the fourth from the unit, is cube, as also are all those which leave out two; [ IX. 8 ] I say that all the rest are also cube.

94 For, since, as the unit is to A , so is A to B , therefore the unit measures A the same number of times as A measures B .

95 But the unit measures A according to the units in it; therefore A also measures B according to the units in itself; therefore A by multiplying itself has made B .

97 But, if a cube number by multiplying itself make some number, the product is cube. [ IX. 3 ]

99 And, since the four numbers A , B , C , D are in continued proportion, and A is cube, D also is cube. [ VIII. 23 ]

100 For the same reason E is also cube, and similarly all the rest are cube. Q. E. D.

PROPOSITION 10.

101 If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be not square, neither will any other be square except the third from the unit and all those which leave out one. And, if the number after the unit be not cube, neither will any other be cube except the fourth from the unit and all those which leave out two.

102 Let there be as many numbers as we please, A , B , C , D , E , F , beginning from an unit and in continued proportion, and let A , the number after the unit, not be square; I say that neither will any other be square except the third from the unit and those which leave out one>.

104 But B is also square; [ IX. 8 ] [therefore B , C have to one another the ratio which a square number has to a square number].

105 And, as B is to C , so is A to B ; therefore A , B have to one another the ratio which a square number has to a square number; [so that A , B are similar plane numbers]. [ VIII. 26 , converse]

106 And B is square; therefore A is also square: which is contrary to the hypothesis.

108 Similarly we can prove that neither is any other of the numbers square except the third from the unit and those which leave out one.

110 I say that neither will any other be cube except the fourth from the unit and those which leave out two.

112 Now C is also cube; for it is fourth from the unit. [ IX. 8 ]

113 And, as C is to D , so is B to C ; therefore B also has to C the ratio which a cube has to a cube.

114 And C is cube; therefore B is also cube. [ VIII. 25 ]

115 And since, as the unit is to A , so is A to B , and the unit measures A according to the units in it, therefore A also measures B according to the units in itself; therefore A by multiplying itself has made the cube number B .

116 But, if a number by multiplying itself make a cube number, it is also itself cube. [ IX. 6 ]

117 Therefore A is also cube: which is contrary to the hypothesis.

119 Similarly we can prove that neither is any other of the numbers cube except the fourth from the unit and those which leave out two. Q. E. D.

PROPOSITION II.

120 If as many numbers as we please beginning from an unit be in continued proportion, the less measures the greater according to some one of the numbers which have place among the proportional numbers.

121 Let there be as many numbers as we please, B , C , D , E , beginning from the unit A and in continued proportion; I say that B , the least of the numbers B , C , D , E , measures E according to some one of the numbers C , D .

122 For since, as the unit A is to B , so is D to E , therefore the unit A measures the number B the same number of times as D measures E ; therefore, alternately, the unit A measures D the same number of times as B measures E . [ VII. 15 ]

123 But the unit A measures D according to the units in it; therefore B also measures E according to the units in D ; so that B the less measures E the greater according to some number of those which have place among the proportional numbers.—

PORISM.

124 And it is manifest that, whatever place the measuring number has, reckoned from the unit, the same place also has the number according to which it measures, reckoned from the number measured, in the direction of the number before it.—

Q. E. D.

PROPOSITION 12.

125 If as many numbers as we please beginning from an unit be in continued proportion, by however many prime numbers the last is measured, the next to the unit will also be measured by the same.

126 Let there be as many numbers as we please, A , B , C , D , beginning from an unit, and in continued proportion; I say that, by however many prime numbers D is measured, A will also be measured by the same.

127 For let D be measured by any prime number E ; I say that E measures A .

128 For suppose it does not; now E is prime, and any prime number is prime to any which it does not measure; [ VII. 29 ] therefore E , A are prime to one another.

129 And, since E measures D , let it measure it according to F , therefore E by multiplying F has made D .

130 Again, since A measures D according to the units in C , [ IX. 11 and Por. ] therefore A by multiplying C has made D .

131 But, further, E has also by multiplying F made D ; therefore the product of A , C is equal to the product of E , F .

132 Therefore, as A is to E , so is F to C . [ VII. 19 ]

133 But A , E are prime, primes are also least, [ VII. 21 ] and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [ VII. 20 ] therefore E measures C .

134 Let it measure it according to G ; therefore E by multiplying G has made C .

135 But, further, by the theorem before this, A has also by multiplying B made C . [ IX. 11 and Por. ]

136 Therefore the product of A , B is equal to the product of E , G .

137 Therefore, as A is to E , so is G to B . [ VII. 19 ]

138 But A , E are prime, primes are also least, [ VII. 21 ] and the least numbers measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent: [ VII. 20 ] therefore E measures B .

139 Let it measure it according to H ; therefore E by multiplying H has made B .

140 But further A has also by multiplying itself made B ; [ IX. 8 ] therefore the product of E , H is equal to the square on A .

141 Therefore, as E is to A , so is A to H . [ VII. 19 ]

142 But A , E are prime, primes are also least, [ VII. 21 ] and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [ VII. 20 ] therefore E measures A , as antecedent antecedent.

143 But, again, it also does not measure it: which is impossible.

146 But numbers composite to one another are measured by some number. [ VII. Def. 14 ]

147 And, since E is by hypothesis prime, and the prime is not measured by any number other than itself, therefore E measures A , E , so that E measures A .

148 [But it also measures D ; therefore E measures A , D .]

149 Similarly we can prove that, by however many prime numbers D is measured, A will also be measured by the same. Q. E. D.

PROPOSITION 13.

150 If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be prime, the greatest will not be measured by any except those which have a place among the proportional numbers .

151 Let there be as many numbers as we please, A , B , C , D , beginning from an unit and in continued proportion, and let A , the number after the unit, be prime; I say that D , the greatest of them, will not be measured by any other number except A , B , C .

152 For, if possible, let it be measured by E , and let E not be the same with any of the numbers A , B , C .

154 For, if E is prime and measures D , it will also measure A [ IX. 12 ], which is prime, though it is not the same with it: which is impossible.

157 But any composite number is measured by some prime number; [ VII. 31 ] therefore E is measured by some prime number.

158 I say next that it will not be measured by any other prime except A .

159 For, if E is measured by another, and E measures D , that other will also measure D ; so that it will also measure A [ IX. 12 ], which is prime, though it is not the same with it: which is impossible.

161 And, since E measures D , let it measure it according to F .

162 I say that F is not the same with any of the numbers A , B , C .

163 For, if F is the same with one of the numbers A , B , C , and measures D according to E , therefore one of the numbers A , B , C also measures D according to E .

164 But one of the numbers A , B , C measures D according to some one of the numbers A , B , C ; [ IX. 11 ] therefore E is also the same with one of the numbers A , B , C : which is contrary to the hypothesis.

165 Therefore F is not the same as any one of the numbers A , B , C .

166 Similarly we can prove that F is measured by A , by proving again that F is not prime.

167 For, if it is, and measures D , it will also measure A [ IX. 12 ], which is prime, though it is not the same with it: which is impossible; therefore F is not prime.

169 But any composite number is measured by some prime number; [ VII. 31 ] therefore F is measured by some prime number.

170 I say next that it will not be measured by any other prime except A .

171 For, if any other prime number measures F , and F measures D , that other will also measure D ; so that it will also measure A [ IX. 12 ], which is prime, though it is not the same with it: which is impossible.

173 And, since E measures D according to F , therefore E by multiplying F has made D .

174 But, further, A has also by multiplying C made D ; [ IX. 11 ] therefore the product of A , C is equal to the product of E , F .

175 Therefore, proportionally, as A is to E , so is F to C . [ VII. 19 ]

178 Similarly, then, we can prove that G is not the same with any of the numbers A , B , and that it is measured by A .

179 And, since F measures C according to G therefore F by multiplying G has made C .

180 But, further, A has also by multiplying B made C ; [ IX. 11 ] therefore the product of A , B is equal to the product of F , G .

181 Therefore, proportionally, as A is to F , so is G to B . [ VII. 19 ]

184 Similarly then we can prove that H is not the same with A .

185 And, since G measures B according to H , therefore G by multiplying H has made B .

186 But further A has also by multiplying itself made B ; [ IX. 8 ] therefore the product of H , G is equal to the square on A .

187 Therefore, as H is to A , so is A to G . [ VII. 19 ]

188 But A measures G ; therefore H also measures A , which is prime, though it is not the same with it: which is absurd.

189 Therefore D the greatest will not be measured by any other number except A , B , C . Q. E. D.

PROPOSITION 14.

190 If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it.

191 For let the number A be the least that is measured by the prime numbers B , C , D ; I say that A will not be measured by any other prime number except B , C , D .

192 For, if possible, let it be measured by the prime number E , and let E not be the same with any one of the numbers B , C , D .

193 Now, since E measures A , let it measure it according to F ; therefore E by multiplying F has made A .

195 But, if two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers; [ VII. 30 ] therefore B , C , D will measure one of the numbers E , F .

196 Now they will not measure E ; for E is prime and not the same with any one of the numbers B , C , D .

197 Therefore they will measure F , which is less than A : which is impossible, for A is by hypothesis the least number measured by B , C , D .

198 Therefore no prime number will measure A except B , C , D . Q. E. D.

PROPOSITION 15.

199 If three numbers in continued proportion be the least of those which have the same ratio with them, any two whatever added together will be prime to the remaining number.

200 Let A , B , C , three numbers in continued proportion, be the least of those which have the same ratio with them; I say that any two of the numbers A , B , C whatever added together are prime to the remaining number, namely A , B to C ; B , C to A ; and further A , C to B .

201 For let two numbers DE , EF , the least of those which have the same ratio with A , B , C , be taken. [ VIII. 2 ]

202 It is then manifest that DE by multiplying itself has made A , and by multiplying EF has made B , and, further, EF by multiplying itself has made C . [ VIII. 2 ]

203 Now, since DE , EF are least, they are prime to one another. [ VII. 22 ]

204 But, if two numbers be prime to one another, their sum is also prime to each; [ VII. 28 ] therefore DF is also prime to each of the numbers DE , EF .

205 But further DE is also prime to EF ; therefore DF , DE are prime to EF .

206 But, if two numbers be prime to any number, their product is also prime to the other; [ VII. 24 ] so that the product of FD , DE is prime to EF ; hence the product of FD , DE is also prime to the square on EF . [ VII. 25 ]

207 But the product of FD , DE is the square on DE together with the product of DE , EF ; [ II. 3 ] therefore the square on DE together with the product of DE , EF is prime to the square on EF .

208 And the square on DE is A , the product of DE , EF is B , and the square on EF is C ; therefore A , B added together are prime to C .

209 Similarly we can prove that B , C added together are prime to A .

210 I say next that A , C added together are also prime to B .

211 For, since DF is prime to each of the numbers DE , EF , the square on DF is also prime to the product of DE , EF . [ VII. 24, 25 ]

212 But the squares on DE , EF together with twice the product of DE , EF are equal to the square on DF ; [ II. 4 ] therefore the squares on DE , EF together with twice the product of DE , EF are prime to the product of DE , EF .

213 Separando , the squares on DE , EF together with once the product of DE , EF are prime to the product of DE , EF .

214 Therefore, separando again, the squares on DE , EF are prime to the product of DE , EF .

215 And the square on DE is A , the product of DE , EF is B , and the square on EF is C .

216 Therefore A , C added together are prime to B . Q. E. D.

PROPOSITION 16.

217 If two numbers be prime to one another, the second will not be to any other number as the first is to the second.

218 For let the two numbers A , B be prime to one another; I say that B is not to any other number as A is to B .

219 For, if possible, as A is to B , so let B be to C .

220 Now A , B are prime, primes are also least, [ VII. 21 ] and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [ VII. 20 ] therefore A measures B as antecedent antecedent.

221 But it also measures itself; therefore A measures A , B which are prime to one another: which is absurd.

222 Therefore B will not be to C , as A is to B . Q. E. D.

PROPOSITION 17.

223 If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the last will not be to any other number as the first to the second.

224 For let there be as many numbers as we please, A , B , C , D , in continued proportion, and let the extremes of them, A , D , be prime to one another; I say that D is not to any other number as A is to B .

225 For, if possible, as A is to B , so let D be to E ; therefore, alternately, as A is to D , so is B to E . [ VII. 13 ]

226 But A , D are prime, primes are also least, [ VII. 21 ] and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent. [ VII. 20 ]

229 Therefore B also measures C ; so that A also measures C .

230 And since, as B is to C , so is C to D , and B measures C , therefore C also measures D .

232 But it also measures itself; therefore A measures A , D which are prime to one another : which is impossible.

233 Therefore D will not be to any other number as A is to B . Q. E. D.

PROPOSITION 18.

234 Given two numbers, to investigate whether it is possible to find a third proportional to them.

235 Let A , B be the given two numbers, and let it be required to investigate whether it is possible to find a third proportional to them.

237 And, if they are prime to one another, it has been proved that it is impossible to find a third proportional to them. [ IX. 16 ]

238 Next, let A , B not be prime to one another, and let B by multiplying itself make C .

240 First, let it measure it according to D ; therefore A by multiplying D has made C .

241 But, further, B has also by multiplying itself made C ; therefore the product of A , D is equal to the square on B .

242 Therefore, as A is to B , so is B to D ; [ VII. 19 ] therefore a third proportional number D has been found to A , B .

243 Next, let A not measure C ; I say that it is impossible to find a third proportional number to A , B .

244 For, if possible, let D , such third proportional, have been found.

245 Therefore the product of A , D is equal to the square on B .

246 But the square on B is C ; therefore the product of A , D is equal to C .

247 Hence A by multiplying D has made C ; therefore A measures C according to D .

248 But, by hypothesis, it also does not measure it: which is absurd.

249 Therefore it is not possible to find a third proportional number to A , B when A does not measure C . Q. E. D.

PROPOSITION 19.

250 Given three numbers, to investigate when it is possible to find a fourth proportional to them.

251 Let A , B , C be the given three numbers, and let it be required to investigate when it is possible to find a fourth proportional to them.

252 Now either they are not in continued proportion, and the extremes of them are prime to one another; or they are in continued proportion, and the extremes of them are not prime to one another; or they are not in continued proportion, nor are the extremes of them prime to one another; or they are in continued proportion, and the extremes of them are prime to one another.

253 If then A , B , C are in continued proportion, and the extremes of them A , C are prime to one another, it has been proved that it is impossible to find a fourth proportional number to them. [ IX. 17 ]

254 Next, let A , B , C not be in continued proportion, the extremes being again prime to one another; I say that in this case also it is impossible to find a fourth proportional to them.

255 For, if possible, let D have been found, so that, as A is to B , so is C to D , and let it be contrived that, as B is to C , so is D to E .

256 Now, since, as A is to B , so is C to D , and, as B is to C , so is D to E , therefore, ex aequali , as A is to C , so is C to E . [ VII. 14 ]

257 But A , C are prime, primes are also least, [ VII. 21 ] and the least numbers measure those which have the same ratio, the antecedent the antecedent and the consequent the consequent. [ VII. 20 ]

259 But it also measures itself; therefore A measures A , C which are prime to one another: which is impossible.

260 Therefore it is not possible to find a fourth proportional to A , B , C .

261 Next, let A , B , C be again in continued proportion, but let A , C not be prime to one another.

262 I say that it is possible to find a fourth proportional to them.

263 For let B by multiplying C make D ; therefore A either measures D or does not measure it.

264 First, let it measure it according to E ; therefore A by multiplying E has made D .

265 But, further, B has also by multiplying C made D ; therefore the product of A , E is equal to the product of B , C ; therefore, proportionally, as A is to B , so is C to E ; [ VII. 19 ] therefore E has been found a fourth proportional to A , B , C .

266 Next, let A not measure D ; I say that it is impossible to find a fourth proportional number to A , B , C .

267 For, if possible, let E have been found; therefore the product of A , E is equal to the product of B , C . [ VII. 19 ]

268 But the product of B , C is D ; therefore the product of A , E is also equal to D .

269 Therefore A by multiplying E has made D ; therefore A measures D according to E , so that A measures D .

271 Therefore it is not possible to find a fourth proportional number to A , B , C when A does not measure D .

272 Next, let A , B , C not be in continued proportion, nor the extremes prime to one another.

274 Similarly then it can be proved that, if A measures D , it is possible to find a fourth proportional to them, but, if it does not measure it, impossible. Q. E. D.

PROPOSITION 20.

275 Prime numbers are more than any assigned multitude of prime numbers.

276 Let A , B , C be the assigned prime numbers; I say that there are more prime numbers than A , B , C .

277 For let the least number measured by A , B , C be taken, and let it be DE ; let the unit DF be added to DE .

279 First, let it be prime; then the prime numbers A , B , C , EF have been found which are more than A , B , C .

280 Next, let EF not be prime; therefore it is measured by some prime number. [ VII. 31 ]

282 I say that G is not the same with any of the numbers A , B , C .

284 Now A , B , C measure DE ; therefore G also will measure DE .

286 Therefore G , being a number, will measure the remainder, the unit DF : which is absurd.

287 Therefore G is not the same with any one of the numbers A , B , C .

289 Therefore the prime numbers A , B , C , G have been found which are more than the assigned multitude of A , B , C . Q. E. D.

PROPOSITION 21.

290 If as many even numbers as we please be added together, the whole is even.

291 For let as many even numbers as we please, AB , BC , CD , DE , be added together; I say that the whole AE is even.

292 For, since each of the numbers AB , BC , CD , DE is even, it has a half part; [ VII. Def. 6 ] so that the whole AE also has a half part.

293 But an even number is that which is divisible into two equal parts; [ id .] therefore AE is even. Q. E. D.

PROPOSITION 22.

294 If as many odd numbers as we please be added together, and their multitude be even, the whole will be even.

295 For let as many odd numbers as we please, AB , BC , CD , DE , even in multitude, be added together; I say that the whole AE is even.

296 For, since each of the numbers AB , BC , CD , DE is odd, if an unit be subtracted from each, each of the remainders will be even; [ VII. Def. 7 ] so that the sum of them will be even. [ IX. 21 ]

298 Therefore the whole AE is also even. [ IX. 21 ] Q. E. D.

PROPOSITION 23.

299 If as many odd numbers as we please be added together, and their multitude be odd, the whole will also be odd.

300 For let as many odd numbers as we please, AB , BC , CD , the multitude of which is odd, be added together; I say that the whole AD is also odd.

301 Let the unit DE be subtracted from CD ; therefore the remainder CE is even. [ VII. Def. 7 ]

302 But CA is also even; [ IX. 22 ] therefore the whole AE is also even. [ IX. 21 ]

PROPOSITION 24.

305 If from an even number an even number be subtracted, the remainder will be even.

306 For from the even number AB let the even number BC be subtracted: I say that the remainder CA is even.

307 For, since AB is even, it has a half part. [ VII. Def. 6 ]

308 For the same reason BC also has a half part; so that the remainder [ CA also has a half part, and] AC is therefore even. Q. E. D.

PROPOSITION 25.

309 If from an even number an odd number be subtracted, the remainder will be odd.

310 For from the even number AB let the odd number BC be subtracted; I say that the remainder CA is odd.

311 For let the unit CD be subtracted from BC ; therefore DB is even. [ VII. Def. 7 ]

312 But AB is also even; therefore the remainder AD is also even. [ IX. 24 ]

313 And CD is an unit; therefore CA is odd. [ VII. Def. 7 ] Q. E. D.

PROPOSITION 26.

314 If from an odd number an odd number be subtracted, the remainder will be even.

315 For from the odd number AB let the odd number BC be subtracted; I say that the remainder CA is even.

316 For, since AB is odd, let the unit BD be subtracted; therefore the remainder AD is even. [ VII. Def. 7 ]

317 For the same reason CD is also even; [ VII. Def. 7 ] so that the remainder CA is also even. [ IX. 24 ] Q. E. D.

PROPOSITION 27.

318 If from an odd number an even number be subtracted, the remainder will be odd.

319 For from the odd number AB let the even number BC be subtracted; I say that the remainder CA is odd.

320 Let the unit AD be subtracted; therefore DB is even. [ VII. Def. 7 ]

321 But BC is also even; therefore the remainder CD is even. [ IX. 24 ]

PROPOSITION 28.

323 If an odd number by multiplying an even number make some number, the product will be even.

324 For let the odd number A by multiplying the even number B make C ; I say that C is even.

325 For, since A by multiplying B has made C , therefore C is made up of as many numbers equal to B as there are units in A . [ VII. Def. 15 ]

326 And B is even; therefore C is made up of even numbers.

327 But, if as many even numbers as we please be added together, the whole is even. [ IX. 21 ]

PROPOSITION 29.

329 If an odd number by multiplying an odd number make some number, the product will be odd.

330 For let the odd number A by multiplying the odd number B make C ; I say that C is odd.

331 For, since A by multiplying B has made C , therefore C is made up of as many numbers equal to B as there are units in A . [ VII. Def. 15 ]

332 And each of the numbers A , B is odd; therefore C is made up of odd numbers the multitude of which is odd.

PROPOSITION 30.

334 If an odd number measure an even number, it will also measure the half of it.

335 For let the odd number A measure the even number B ; I say that it will also measure the half of it.

336 For, since A measures B , let it measure it according to C ; I say that C is not odd.

338 Then, since A measures B according to C , therefore A by multiplying C has made B .

339 Therefore B is made up of odd numbers the multitude of which is odd.

340 Therefore B is odd: [ IX. 23 ] which is absurd, for by hypothesis it is even.

343 For this reason then it also measures the half of it. Q. E. D.

PROPOSITION 31.

344 If an odd number be prime to any number, it will also be prime to the double of it.

345 For let the odd number A be prime to any number B , and let C be double of B ; I say that A is prime to C .

346 For, if they are not prime to one another, some number will measure them.

349 And since D which is odd measures C , and C is even, therefore [ D ] will measure the half of C also. [ IX. 30 ]

351 But it also measures A ; therefore D measures A , B which are prime to one another: which is impossible.

PROPOSITION 32.

354 Each of the numbers which are continually doubled beginning from a dyad is even-times even only.

355 For let as many numbers as we please, B , C , D , have been continually doubled beginning from the dyad A ; I say that B , C , D are eventimes even only.

356 Now that each of the numbers B , C , D is even-times even is manifest; for it is doubled from a dyad.

359 Since then as many numbers as we please beginning from an unit are in continued proportion, and the number A after the unit is prime, therefore D , the greatest of the numbers A , B , C , D , will not be measured by any other number except A , B , C . [ IX. 13 ]

360 And each of the numbers A , B , C is even; therefore D is even-times even only. [ VII. Def. 8 ]

361 Similarly we can prove that each of the numbers B , C is even-times even only. Q. E. D.

PROPOSITION 33.

362 If a number have its half odd, it is even-times odd only.

363 For let the number A have its half odd; I say that A is even-times odd only.

364 Now that it is even-times odd is manifest; for the half of it, being odd, measures it an even number of times. [ VII. Def. 9 ]

366 For, if A is even-times even also, it will be measured by an even number according to an even number; [ VII. Def. 8 ] so that the half of it will also be measured by an even number though it is odd: which is absurd.

PROPOSITION 34.

368 If a number neither be one of those which are continually doubled from a dyad, nor have its half odd, it is both eventimes even and even-times odd .

369 For let the number A neither be one of those doubled from a dyad, nor have its half odd; I say that A is both even-times even and even-times odd.

370 Now that A is even-times even is manifest; for it has not its half odd. [ VII. Def. 8 ]

372 For, if we bisect A , then bisect its half, and do this continually, we shall come upon some odd number which will measure A according to an even number.

373 For, if not, we shall come upon a dyad, and A will be among those which are doubled from a dyad: which is contrary to the hypothesis.

376 Therefore A is both even-times even and even-times odd. Q. E. D.

PROPOSITION 35.

377 If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last numbers equal to the first, then, as the excess of the second is to the first, so will the excess of the last be to all those before it.

378 Let there be as many numbers as we please in continued proportion, A , BC , D , EF , beginning from A as least, and let there be subtracted from BC and EF the numbers BG , FH , each equal to A ; I say that, as GC is to A , so is EH to A , BC , D .

379 For let FK be made equal to BC , and FL equal to D .

380 Then, since FK is equal to BC , and of these the part FH is equal to the part BG , therefore the remainder HK is equal to the remainder GC .

381 And since, as EF is to D , so is D to BC , and BC to A , while D is equal to FL , BC to FK , and A to FH , therefore, as EF is to FL , so is LF to FK , and FK to FH .

382 Separando , as EL is to LF , so is LK to FK , and KH to FH . [ VII. 11, 13 ]

383 Therefore also, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents; [ VII. 12 ] therefore, as KH is to FH , so are EL , LK . KH to LF , FK , HF .

384 But KH is equal to CG , FH to A , and LF , FK , HF to D , BC , A ; therefore, as CG is to A , so is EH to D , BC , A .

385 Therefore, as the excess of the second is to the first, so is the excess of the last to all those before it. Q. E. D.

PROPOSITION 36.

386 If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect.

387 For let as many numbers as we please, A , B , C , D , beginning from an unit be set out in double proportion, until the sum of all becomes prime, let E be equal to the sum, and let E by multiplying D make FG ; I say that FG is perfect.

388 For, however many A , B , C , D are in multitude, let so many E , HK , L , M be taken in double proportion beginning from E ; therefore, ex aequali , as A is to D , so is E to M . [ VII. 14 ]

389 Therefore the product of E , D is equal to the product of A , M . [ VII. 19 ]

390 And the product of E , D is FG ; therefore the product of A , M is also FG .

391 Therefore A by multiplying M has made FG ; therefore M measures FG according to the units in A .

393 But M , L , HK , E are continuously double of each other; therefore E , HK , L , M , FG are continuously proportional in double proportion.

394 Now let there be subtracted from the second HK and the last FG the numbers HN , FO , each equal to the first E ; therefore, as the excess of the second is to the first, so is the excess of the last to all those before it. [ IX. 35 ]

395 Therefore, as NK is to E , so is OG to M , L , KH , E .

396 And NK is equal to E ; therefore OG is also equal to M , L , HK , E .

397 But FO is also equal to E , and E is equal to A , B , C , D and the unit.

398 Therefore the whole FG is equal to E , HK , L , M and A , B , C , D and the unit; and it is measured by them.

399 I say also that FG will not be measured by any other number except A , B , C , D , E , HK , L , M and the unit.

400 For, if possible, let some number P measure FG , and let P not be the same with any of the numbers A , B , C , D , E , HK , L , M .