Elements, Book 7

By Euclid

Edition: 0.0.0-dev | March 03, 2014

Authority: SCTA

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BOOK VII.

DEFINITIONS.

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1 An unit is that by virtue of which each of the things that exist is called one.

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3 A number is a part of a number, the less of the greater, when it measures the greater;

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5 The greater number is a multiple of the less when it is measured by the less.

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6 An even number is that which is divisible into two equal parts.

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7 An odd number is that which is not divisible into two equal parts, or that which differs by an unit from an even number.

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8 An even-times even number is that which is measured by an even number according to an even number.

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9 An even-times odd number is that which is measured by an even number according to an odd number.

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10 An odd-times odd number is that which is measured by an odd number according to an odd number.

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11 A prime number is that which is measured by an unit alone.

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12 Numbers prime to one another are those which are measured by an unit alone as a common measure.

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13 A composite number is that which is measured by some number.

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14 Numbers composite to one another are those which are measured by some number as a common measure.

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15 A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.

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16 And, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.

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17 And, when three numbers having multiplied one another make some number, the number so produced is solid , and its sides are the numbers which have multiplied one another.

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18 A square number is equal multiplied by equal, or a number which is contained by two equal numbers.

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19 And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.

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20 Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.

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21 Similar plane and solid numbers are those which have their sides proportional.

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22 A perfect number is that which is equal to its own parts.

BOOK VII. PROPOSITIONS.

PROPOSITION I.

23 Two unequal numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit is left, the original numbers will be prime to one another.

24 For, the less of two unequal numbers AB , CD being continually subtracted from the greater, let the number which is left never measure the one before it until an unit is left; I say that AB , CD are prime to one another, that is, that an unit alone measures AB , CD .

25 For, if AB , CD are not prime to one another, some number will measure them.

26 Let a number measure them, and let it be E ; let CD , measuring BF , leave FA less than itself, let AF , measuring DG , leave GC less than itself, and let GC , measuring FH , leave an unit HA .

27 Since, then, E measures CD , and CD measures BF , therefore E also measures BF .

28 But it also measures the whole BA ; therefore it will also measure the remainder AF .

30 But it also measures the whole DC therefore it will also measure the remainder CG .

32 But it also measures the whole FA ; therefore it will also measure the remainder, the unit AH , though it is a number: which is impossible.

33 Therefore no number will measure the numbers AB , CD ; therefore AB , CD are prime to one another. [ VII. Def. 12 ] Q. E. D.

PROPOSITION 2.

34 Given two numbers not prime to one another, to find their greatest common measure.

35 Let AB , CD be the two given numbers not prime to one another.

36 Thus it is required to find the greatest common measure of AB , CD .

37 If now CD measures AB —and it also measures itself— CD is a common measure of CD , AB .

38 And it is manifest that it is also the greatest; for no greater number than CD will measure CD .

39 But, if CD does not measure AB , then, the less of the numbers AB , CD being continually subtracted from the greater, some number will be left which will measure the one before it.

40 For an unit will not be left; otherwise AB , CD will be prime to one another [ VII. 1 ], which is contrary to the hypothesis.

41 Therefore some number will be left which will measure the one before it.

42 Now let CD , measuring BE , leave EA less than itself, let EA , measuring DF , leave FC less than itself, and let CF measure AE .

43 Since then, CF measures AE , and AE measures DF , therefore CF will also measure DF .

44 But it also measures itself; therefore it will also measure the whole CD .

45 But CD measures BE ; therefore CF also measures BE .

46 But it also measures EA ; therefore it will also measure the whole BA .

47 But it also measures CD ; therefore CF measures AB , CD .

50 For, if CF is not the greatest common measure of AB , CD , some number which is greater than CF will measure the numbers AB , CD .

52 Now, since G measures CD , while CD measures BE , G also measures BE .

53 But it also measures the whole BA ; therefore it will also measure the remainder AE .

54 But AE measures DF ; therefore G will also measure DF .

55 But it also measures the whole DC ; therefore it will also measure the remainder CF , that is, the greater will measure the less: which is impossible.

56 Therefore no number which is greater than CF will measure the numbers AB , CD ; therefore CF is the greatest common measure of AB , CD .

PORISM.

57 From this it is manifest that, if a number measure two numbers, it will also measure their greatest common measure.

Q. E. D.

PROPOSITION 3.

58 Given three numbers not prime to one another, to find their greatest common measure.

59 Let A , B , C be the three given numbers not prime to one another; thus it is required to find the greatest common measure of A , B , C .

60 For let the greatest common measure, D , of the two numbers A , B be taken; [ VII. 2 ] then D either measures, or does not measure, C .

62 But it measures A , B also; therefore D measures A , B , C ; therefore D is a common measure of A , B , C .

64 For, if D is not the greatest common measure of A , B , C , some number which is greater than D will measure the numbers A , B , C .

66 Since then E measures A , B , C , it will also measure A , B ; therefore it will also measure the greatest common measure of A , B . [ VII. 2, Por. ]

67 But the greatest common measure of A , B is D ; therefore E measures D , the greater the less: which is impossible.

68 Therefore no number which is greater than D will measure the numbers A , B , C ; therefore D is the greatest common measure of A , B , C .

69 Next, let D not measure C ; I say first that C , D are not prime to one another.

70 For, since A , B , C are not prime to one another, some number will measure them.

71 Now that which measures A , B , C will also measure A , B , and will measure D , the greatest common measure of A , B . [ VII. 2, Por. ]

72 But it measures C also; therefore some number will measure the numbers D , C ; therefore D , C are not prime to one another.

73 Let then their greatest common measure E be taken. [ VII. 2 ]

74 Then, since E measures D , and D measures A , B , therefore E also measures A , B .

75 But it measures C also; therefore E measures A , B , C ; therefore E is a common measure of A , B , C .

77 For, if E is not the greatest common measure of A , B , C , some number which is greater than E will measure the numbers A , B , C .

79 Now, since F measures A , B , C , it also measures A , B ; therefore it will also measure the greatest common measure of A , B . [ VII. 2, Por. ]

80 But the greatest common measure of A , B is D ; therefore F measures D .

81 And it measures C also; therefore F measures D , C ; therefore it will also measure the greatest common measure of D , C . [ VII. 2, Por. ]

82 But the greatest common measure of D , C is E ; therefore F measures E , the greater the less: which is impossible.

83 Therefore no number which is greater than E will measure the numbers A , B , C ; therefore E is the greatest common measure of A , B , C . Q. E. D.

PROPOSITION 4.

84 Any number is either a part or parts of any number, the less of the greater.

85 Let A , BC be two numbers, and let BC be the less; I say that BC is either a part, or parts, of A .

88 Then, if BC be divided into the units in it, each unit of those in BC will be some part of A ; so that BC is parts of A .

89 Next let A , BC not be prime to one another; then BC either measures, or does not measure, A .

91 But, if not, let the greatest common measure D of A , BC be taken; [ VII. 2 ] and let BC be divided into the numbers equal to D , namely BE , EF , FC .

93 But D is equal to each of the numbers BE , EF , FC ; therefore each of the numbers BE , EF , FC is also a part of A ; so that BC is parts of A .

PROPOSITION 5.

95 If a number be a part of a number, and another be the same part of another, the sum will also be the same part of the sum that the one is of the one.

96 For let the number A be a part of BC , and another, D , the same part of another EF that A is of BC ; I say that the sum of A , D is also the same part of the sum of BC , EF that A is of BC .

97 For since, whatever part A is of BC , D is also the same part of EF , therefore, as many numbers as there are in BC equal to A , so many numbers are there also in EF equal to D .

98 Let BC be divided into the numbers equal to A , namely BG , GC , and EF into the numbers equal to D , namely EH , HF ; then the multitude of BG , GC will be equal to the multitude of EH , HF .

99 And, since BG is equal to A , and EH to D , therefore BG , EH are also equal to A , D .

100 For the same reason GC , HF are also equal to A , D .

101 Therefore, as many numbers as there are in BC equal to A , so many are there also in BC , EF equal to A , D .

102 Therefore, whatever multiple BC is of A , the same multiple also is the sum of BC , EF of the sum of A , D .

103 Therefore, whatever part A is of BC , the same part also is the sum of A , D of the sum of BC , EF . Q. E. D.

PROPOSITION 6.

104 If a number be parts of a number, and another be the same parts of another, the sum will also be the same parts of the sum that the one is of the one.

105 For let the number AB be parts of the number C , and another, DE , the same parts of another, F , that AB is of C ; I say that the sum of AB , DE is also the same parts of the sum of C , F that AB is of C .

106 For since, whatever parts AB is of C , DE is also the same parts of F , therefore, as many parts of C as there are in AB , so many parts of F are there also in DE .

107 Let AB be divided into the parts of C , namely AG , GB , and DE into the parts of F , namely DH , HE ; thus the multitude of AG , GB will be equal to the multitude of DH , HE .

108 And since, whatever part AG is of C , the same part is DH of F also, therefore, whatever part AG is of C , the same part also is the sum of AG , DH of the sum of C , F . [ VII. 5 ]

109 For the same reason, whatever part GB is of C , the same part also is the sum of GB , HE of the sum of C , F .

110 Therefore, whatever parts AB is of C , the same parts also is the sum of AB , DE of the sum of C , F . Q. E. D.

PROPOSITION 7.

111 If a number be that part of a number, which a number subtracted is of a number subtracted, the remainder will also be the same part of the remainder that the whole is of the whole.

112 For let the number AB be that part of the number CD which AE subtracted is of CF subtracted; I say that the remainder EB is also the same part of the remainder FD that the whole AB is of the whole CD .

113 For, whatever part AE is of CF , the same part also let EB be of CG .

114 Now since, whatever part AE is of CF , the same part also is EB of CG , therefore, whatever part AE is of CF , the same part also is AB of GF . [ VII. 5 ]

115 But, whatever part AE is of CF , the same part also, by hypothesis, is AB of CD ; therefore, whatever part AB is of GF , the same part is it of CD also; therefore GF is equal to CD .

116 Let CF be subtracted from each; therefore the remainder GC is equal to the remainder FD .

117 Now since, whatever part AE is of CF , the same part also is EB of GC , while GC is equal to FD , therefore, whatever part AE is of CF , the same part also is EB of FD .

118 But, whatever part AE is of CF , the same part also is AB of CD ; therefore also the remainder EB is the same part of the remainder FD that the whole AB is of the whole CD . Q. E. D.

PROPOSITION 8.

119 If a number be the same parts of a number that a number subtracted is of a number subtracted, the remainder will also be the same parts of the remainder that the whole is of the whole.

120 For let the number AB be the same parts of the number CD that AE subtracted is of CF subtracted; I say that the remainder EB is also the same parts of the remainder FD that the whole AB is of the whole CD .

122 Therefore, whatever parts GH is of CD , the same parts also is AE of CF .

123 Let GH be divided into the parts of CD , namely GK , KH , and AE into the parts of CF , namely AL , LE ; thus the multitude of GK , KH will be equal to the multitude of AL , LE .

124 Now since, whatever part GK is of CD , the same part also is AL of CF , while. CD is greater than CF , therefore GK is also greater than AL .

126 Therefore, whatever part GK is of CD , the same part also is GM of CF ; therefore also the remainder MK is the same part of the remainder FD that the whole GK is of the whole CD . [ VII. 7 ]

127 Again, since, whatever part KH is of CD , the same part also is EL of CF , while CD is greater than CF , therefore HK is also greater than EL .

129 Therefore, whatever part KH is of CD , the same part also is KN of CF ; therefore also the remainder NH is the same part of the remainder FD that the whole KH is of the whole CD . [ VII. 7 ]

130 But the remainder MK was also proved to be the same part of the remainder FD that the whole GK is of the whole CD ; therefore also the sum of MK , NH is the same parts of DF that the whole HG is of the whole CD .

131 But the sum of MK , NH is equal to EB , and HG is equal to BA ; therefore the remainder EB is the same parts of the remainder FD that the whole AB is of the whole CD . Q. E. D.

PROPOSITION 9.

132 If a number be a part of a number, and another be the same part of another, alternately also, whatever part or parts the first is of the third, the same part, or the same parts, will the second also be of the fourth.

133 For let the number A be a part of the number BC , and another, D , the same part of another, EF , that A is of BC ; I say that, alternately also, whatever part or parts A is of D , the same part or parts is BC of EF also.

134 For since, whatever part A is of BC , the same part also is D of EF , therefore, as many numbers as there are in BC equal to A , so many also are there in EF equal to D .

135 Let BC be divided into the numbers equal to A , namely BG , GC , and EF into those equal to D , namely EH , HF ; thus the multitude of BG , GC will be equal to the multitude of EH , HF .

136 Now, since the numbers BG , GC are equal to one another, and the numbers EH , HF are also equal to one another, while the multitude of BG , GC is equal to the multitude of EH , HF , therefore, whatever part or parts BG is of EH , the same part or the same parts is GC of HF also; so that, in addition, whatever part or parts BG is of EH , the same part also, or the same parts, is the sum BC of the sum EF . [ VII. 5, 6 ]

137 But BG is equal to A , and EH to D ; therefore, whatever part or parts A is of D , the same part or the same parts is BC of EF also. Q. E. D.

PROPOSITION 10.

138 If a number be parts of a number, and another be the same parts of another, alternately also, whatever parts or part the first is of the third, the same parts or the same part will the second also be of the fourth.

139 For let the number AB be parts of the number C , and another, DE , the same parts of another, F ; I say that, alternately also, whatever parts or part AB is of DE , the same parts or the same part is C of F also.

140 For since, whatever parts AB is of C , the same parts also is DE of F , therefore, as many parts of C as there are in AB , so many parts also of F are there in DE .

141 Let AB be divided into the parts of C , namely AG , GB , and DE into the parts of F , namely DH , HE ; thus the multitude of AG , GB will be equal to the multitude of DH , HE .

142 Now since, whatever part AG is of C , the same part also is DH of F , alternately also, whatever part or parts AG is of DH , the same part or the same parts is C of F also. [ VII. 9 ]

143 For the same reason also, whatever part or parts GB is of HE , the same part or the same parts is C of F also; so that, in addition, whatever parts or part AB is of DE , the same parts also, or the same part, is C of F . [ VII. 5, 6 ] Q. E. D.

PROPOSITION 11.

144 If, as whole is to whole, so is a number subtracted to a number subtracted, the remainder will also be to the remainder as whole to whole.

145 As the whole AB is to the whole CD , so let AE subtracted be to CF subtracted; I say that the remainder EB is also to the remainder FD as the whole AB to the whole CD .

146 Since, as AB is to CD , so is AE to CF , whatever part or parts AB is of CD , the same part or the same parts is AE of CF also; [ VII. Def. 20 ]

147 Therefore also the remainder EB is the same part or parts of FD that AB is of CD . [ VII. 7, 8 ]

148 Therefore, as EB is to FD , so is AB to CD . [ VII. Def. 20 ] Q. E. D.

PROPOSITION 12.

149 If there be as many numbers as we please in proportion, then, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents.

150 Let A , B , C , D be as many numbers as we please in proportion, so that, as A is to B , so is C to D ; I say that, as A is to B , so are A , C to B , D .

151 For since, as A is to B , so is C to D , whatever part or parts A is of B , the same part or parts is C of D also. [ VII. Def. 20 ]

152 Therefore also the sum of A , C is the same part or the same parts of the sum of B , D that A is of B . [ VII. 5, 6 ]

153 Therefore, as A is to B , so are A , C to B , D . [ VII. Def. 20 ]

PROPOSITION 13.

154 If four numbers be proportional, they will also be proportional alternately.

155 Let the four numbers A , B , C , D be proportional, so that, as A is to B , so is C to D ; I say that they will also be proportional alternately, so that, as A is to C , so will B be to D .

156 For since, as A is to B , so is C to D , therefore, whatever part or parts A is of B , the same part or the same parts is C of D also. [ VII. Def. 20 ]

157 Therefore, alternately, whatever part or parts A is of C , the same part or the same parts is B of D also. [ VII. 10 ]

158 Therefore, as A is to C , so is B to D . [ VII. Def. 20 ] Q. E. D.

PROPOSITION 14.

159 If there be as many numbers as we please, and others equal to them in multitude, which taken two and two are in the same ratio, they will also be in the same ratio ex aequali.

160 Let there be as many numbers as we please A , B , C , and others equal to them in multitude D , E , F , which taken two and two are in the same ratio, so that, as A is to B , so is D to E , and, as B is to C , so is E to F ; I say that, ex aequali , as A is to C , so also is D to F .

161 For, since, as A is to B , so is D to E , therefore, alternately, as A is to D , so is B to E . [ VII. 13 ]

162 Again, since, as B is to C , so is E to F , therefore, alternately, as B is to E , so is C to F . [ VII. 13 ]

163 But, as B is to E , so is A to D ; therefore also, as A is to D , so is C to F .

164 Therefore, alternately, as A is to C , so is D to F . [ id .]

PROPOSITION 15.

165 If an unit measure any number, and another number measure any other number the same number of times, alternately also, the unit will measure the third number the same number of times that the second measures the fourth.

166 For let the unit A measure any number BC , and let another number D measure any other number EF the same number of times; I say that, alternately also, the unit A measures the number D the same number of times that BC measures EF .

167 For, since the unit A measures the number BC the same number of times that D measures EF , therefore, as many units as there are in BC , so many numbers equal to D are there in EF also.

168 Let BC be divided into the units in it, BG , GH , HC , and EF into the numbers EK , KL , LF equal to D .

169 Thus the multitude of BG , GH , HC will be equal to the multitude of EK , KL , LF .

170 And, since the units BG , GH , HC are equal to one another, and the numbers EK , KL , LF are also equal to one another, while the multitude of the units BG , GH , HC is equal to the multitude of the numbers EK , KL , LF , therefore, as the unit BG is to the number EK , so will the unit GH be to the number KL , and the unit HC to the number LF .

171 Therefore also, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents; [ VII. 12 ] therefore, as the unit BG is to the number EK , so is BC to EF .

172 But the unit BG is equal to the unit A , and the number EK to the number D .

173 Therefore, as the unit A is to the number D , so is BC to EF .

174 Therefore the unit A measures the number D the same number of times that BC measures EF . Q. E. D.

PROPOSITION 16.

175 If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.

176 Let A , B be two numbers, and let A by multiplying B make C , and B by multiplying A make D ; I say that C is equal to D .

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

178 But the unit E also measures the number A according to the units in it; therefore the unit E measures A the same number of times that B measures C .

179 Therefore, alternately, the unit E measures the number B the same number of times that A measures C . [ VII. 15 ]

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

181 But the unit E also measures B according to the units in it; therefore the unit E measures the number B the same number of times that A measures D .

182 But the unit E measured the number B the same number of times that A measures C ; therefore A measures each of the numbers C , D the same number of times.

PROPOSITION 17.

184 If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.

185 For let the number A by multiplying the two numbers B , C make D , E ; I say that, as B is to C , so is D to E .

186 For, since A by multiplying B has made D , therefore B measures D according to the units in A .

187 But the unit F also measures the number A according to the units in it; therefore the unit F measures the number A the same number of times that B measures D .

188 Therefore, as the unit F is to the number A , so is B to D . [ VII. Def. 20 ]

189 For the same reason, as the unit F is to the number A , so also is C to E ; therefore also, as B is to D , so is C to E .

190 Therefore, alternately, as B is to C , so is D to E . [ VII. 13 ] Q. E. D.

PROPOSITION 18.

191 If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.

192 For let two numbers A , B by multiplying any number C make D , E ; I say that, as A is to B , so is D to E .

193 For, since A by multiplying C has made D , therefore also C by multiplying A has made D . [ VII. 16 ] For the same reason also C by multiplying B has made E .

194 Therefore the number C by multiplying the two numbers A , B has made D , E .

195 Therefore, as A is to B , so is D to E . [ VII. 17 ]

PROPOSITION 19.

196 If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.

197 Let A , B , C , D be four numbers in proportion, so that, as A is to B , so is C to D ; and let A by multiplying D make E , and let B by multiplying C make F ; I say that E is equal to F .

199 Since, then, A by multiplying C has made G , and by multiplying D has made E , the number A by multiplying the two numbers C , D has made G , E .

200 Therefore, as C is to D , so is G to E . [ VII. 17 ]

201 But, as C is to D , so is A to B ; therefore also, as A is to B , so is G to E .

202 Again, since A by multiplying C has made G , but, further, B has also by multiplying C made F , the two numbers A , B by multiplying a certain number C have made G , F .

203 Therefore, as A is to B , so is G to F . [ VII. 18 ]

204 But further, as A is to B , so is G to E also; therefore also, as G is to E , so is G to F .

205 Therefore G has to each of the numbers E , F the same ratio; therefore E is equal to F . [cf. V. 9 ]

206 Again, let E be equal to F ; I say that, as A is to B , so is C to D .

207 For, with the same construction, since E is equal to F , therefore, as G is to E , so is G to F . [cf. V. 7 ]

208 But, as G is to E , so is C to D , [ VII. 17 ] and, as G is to F , so is A to B . [ VII. 18 ]

209 Therefore also, as A is to B , so is C to D . Q. E. D.

PROPOSITION 20.

210 The least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less.

211 For let CD , EF be the least numbers of those which have the same ratio with A , B ; I say that CD measures A the same number of times that EF measures B .

213 For, if possible, let it be so; therefore EF is also the same parts of B that CD is of A . [ VII. 13 and Def. 20 ]

214 Therefore, as many parts of A as there are in CD , so many parts of B are there also in EF .

215 Let CD be divided into the parts of A , namely CG , GD , and EF into the parts of B , namely EH , HF ; thus the multitude of CG , GD will be equal to the multitude of EH , HF .

216 Now, since the numbers CG , GD are equal to one another, and the numbers EH , HF are also equal to one another, while the multitude of CG , GD is equal to the multitude of EH , HF , therefore, as CG is to EH , so is GD to HF .

217 Therefore also, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents. [ VII. 12 ]

219 Therefore CG , EH are in the same ratio with CD , EF , being less than they: which is impossible, for by hypothesis CD , EF are the least numbers of those which have the same ratio with them.

220 Therefore CD is not parts of A ; therefore it is a part of it. [ VII. 4 ]

221 And EF is the same part of B that CD is of A ; [ VII. 13 and Def. 20 ] therefore CD measures A the same number of times that EF measures B . Q. E. D.

PROPOSITION 21.

222 Numbers prime to one another are the least of those which have the same ratio with them.

223 Let A , B be numbers prime to one another; I say that A , B are the least of those which have the same ratio with them.

224 For, if not, there will be some numbers less than A , B which are in the same ratio with A , B .

226 Since, then, the least numbers of those which have the same ratio measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent, [ VII. 20 ] therefore C measures A the same number of times that D measures B .

227 Now, as many times as C measures A , so many units let there be in E .

228 Therefore D also measures B according to the units in E .

229 And, since C measures A according to the units in E , therefore E also measures A according to the units in C . [ VII. 16 ]

230 For the same reason E also measures B according to the units in D . [ VII. 16 ]

231 Therefore E measures A , B which are prime to one another: which is impossible. [ VII. Def. 12 ]

232 Therefore there will be no numbers less than A , B which are in the same ratio with A , B .

233 Therefore A , B are the least of those which have the same ratio with them. Q. E. D.

PROPOSITION 22.

234 The least numbers of those which have the same ratio with them are prime to one another.

235 Let A , B be the least numbers of those which have the same ratio with them; I say that A , B are prime to one another.

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

238 And, as many times as C measures A , so many units let there be in D , and, as many times as C measures B , so many units let there be in E

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

240 For the same reason also C by multiplying E has made B .

241 Thus the number C by multiplying the two numbers D , E has made A , B ; therefore, as D is to E , so is A to B ; [ VII. 17 ] therefore D , E are in the same ratio with A , B , being less than they: which is impossible.

242 Therefore no number will measure the numbers A , B .

PROPOSITION 23.

244 If two number be prime to one another, the number which measures the one of them will be prime to the remaining number.

245 Let A , B be two numbers prime to one another, and let any number C measure A ; I say that C , B are also prime to one another.

246 For, if C , B are not prime to one another, some number will measure C , B .

248 Since D measures C , and C measures A , therefore D also measures A .

249 But it also measures B ; therefore D measures A , B which are prime to one another: which is impossible. [ VII. Def. 12 ]

250 Therefore no number will measure the numbers C , B .

PROPOSITION 24.

252 If two numbers be prime to any number, their product also will be prime to the same.

253 For let the two numbers A , B be prime to any number C , and let A by multiplying B make D ; I say that C , D are prime to one another.

254 For, if C , D are not prime to one another, some number will measure C , D .

256 Now, since C , A are prime to one another, and a certain number E measures C , therefore A , E are prime to one another. [ VII. 23 ]

257 As many times, then, as E measures D , so many units let there be in F ; therefore F also measures D according to the units in E . [ VII. 16 ]

258 Therefore E by multiplying F has made D . [ VII. Def. 15 ]

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

260 But, if the product of the extremes be equal to that of the means, the four numbers are proportional; [ VII. 19 ] therefore, as E is to A , so is B to F .

261 But A , E are prime to one another, numbers which are prime to one another are also the least of those which have the same ratio, [ VII. 21 ] and the least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [ VII. 20 ] therefore E measures B .

262 But it also measures C ; therefore E measures B , C which are prime to one another: which is impossible. [ VII. Def. 12 ]

263 Therefore no number will measure the numbers C , D .

PROPOSITION 25.

265 If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.

266 Let A , B be two numbers prime to one another, and let A by multiplying itself make C : I say that B , C are prime to one another.

268 Since A , B are prime to one another, and A is equal to D , therefore D , B are also prime to one another.

269 Therefore each of the two numbers D , A is prime to B ; therefore the product of D , A will also be prime to B . [ VII. 24 ]

270 But the number which is the product of D , A is C .

PROPOSITION 26.

272 If two numbers be prime to two numbers, both to each, their products also will be prime to one another.

273 For let the two numbers A , B be prime to the two numbers C , D ; both to each, and let A by multiplying B make E , and let C by multiplying D make F ; I say that E , F are prime to one another.

274 For, since each of the numbers A , B is prime to C , therefore the product of A , B will also be prime to C . [ VII. 24 ]

275 But the product of A , B is E ; therefore E , C are prime to one another.

276 For the same reason E , D are also prime to one another.

277 Therefore each of the numbers C , D is prime to E .

278 Therefore the product of C , D will also be prime to E . [ VII. 24 ]

PROPOSITION 27.

281 If two numbers be prime to one another, and each by multiplying itself make a certain number, the products will be prime to one another; and, if the original numbers by multiplying the products make certain numbers, the latter will also be prime to one another [ and this is always the case with the extremes ].

282 Let A , B be two numbers prime to one another, let A by multiplying itself make C , and by multiplying C make D , and let B by multiplying itself make E , and by multiplying E make F ; I say that both C , E and D , F are prime to one another.

283 For, since A , B are prime to one another, and A by multiplying itself has made C , therefore C , B are prime to one another. [ VII. 25 ]

284 Since then C , B are prime to one another, and B by multiplying itself has made E , therefore C , E are prime to one another. [ id .]

285 Again, since A , B are prime to one another, and B by multiplying itself has made E , therefore A , E are prime to one another. [ id .]

286 Since then the two numbers A , C are prime to the two numbers B , E , both to each, therefore also the product of A , C is prime to the product of B , E . [ VII. 26 ]

287 And the product of A , C is D , and the product of B , E is F .

PROPOSITION 28.

289 If two numbers be prime to one another, the sum will also be prime to each of them; and, if the sum of two numbers be prime to any one of them, the original numbers will also be prime to one another.

290 For let two numbers AB , BC prime to one another be added; I say that the sum AC is also prime to each of the numbers AB , BC .

291 For, if CA , AB are not prime to one another, some number will measure CA , AB .

293 Since then D measures CA , AB , therefore it will also measure the remainder BC .

294 But it also measures BA ; therefore D measures AB , BC which are prime to one another: which is impossible. [ VII. Def. 12 ]

295 Therefore no number will measure the numbers CA , AB ; therefore CA , AB are prime to one another.

296 For the same reason AC , CB are also prime to one another.

297 Therefore CA is prime to each of the numbers AB , BC .

298 Again, let CA , AB be prime to one another; I say that AB , BC are also prime to one another.

299 For, if AB , BC are not prime to one another, some number will measure AB , BC .

301 Now, since D measures each of the numbers AB , BC , it will also measure the whole CA .

302 But it also measures AB ; therefore D measures CA , AB which are prime to one another: which is impossible. [ VII. Def. 12 ]

303 Therefore no number will measure the numbers AB , BC .

304 Therefore AB , BC are prime to one another. Q. E. D.

PROPOSITION 29.

305 Any prime number is prime to any number which it does not measure.

306 Let A be a prime number, and let it not measure B ; I say that B , A are prime to one another.

307 For, if B , A are not prime to one another, some number will measure them.

309 Since C measures B , and A does not measure B , therefore C is not the same with A .

310 Now, since C measures B , A , therefore it also measures A which is prime, though it is not the same with it: which is impossible.

PROPOSITION 30.

313 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.

314 For let the two numbers A , B by multiplying one another make C , and let any prime number D measure C ; I say that D measures one of the numbers A , B .

316 Now D is prime; therefore A , D are prime to one another. [ VII. 29 ]

317 And, as many times as D measures C , so many units let there be in E .

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

319 Further, A by multiplying B has also made C ; therefore the product of D , E is equal to the product of A , B .

320 Therefore, as D is to A , so is B to E . [ VII. 19 ]

321 But D , A are prime to one another, primes are also least, [ VII. 21 ] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [ VII. 20 ] therefore D measures B .

322 Similarly we can also show that, if D do not measure B , it will measure A .

323 Therefore D measures one of the numbers A , B . Q. E. D.

PROPOSITION 31.

324 Any composite number is measured by some prime number.

325 Let A be a composite number; I say that A is measured by some prime number.

326 For, since A is composite, some number will measure it.

328 Now, if B is prime, what was enjoined will have been done.

329 But if it is composite, some number will measure it.

331 Then, since C measures B , and B measures A , therefore C also measures A .

332 And, if C is prime, what was enjoined will have been done.

333 But if it is composite, some number will measure it.

334 Thus, if the investigation be continued in this way, some prime number will be found which will measure the number before it, which will also measure A .

335 For, if it is not found, an infinite series of numbers will measure the number A , each of which is less than the other: which is impossible in numbers.

336 Therefore some prime number will be found which will measure the one before it, which will also measure A .

337 Therefore any composite number is measured by some prime number.

PROPOSITION 32.

338 Any number either is prime or is measured by some prime number.

339 Let A be a number; I say that A either is prime or is measured by some prime number.

340 If now A is prime, that which was enjoined will have been done.

341 But if it is composite, some prime number will measure it. [ VII. 31 ]

342 Therefore any number either is prime or is measured by some prime number. Q. E. D.

PROPOSITION 33.

343 Given as many numbers as we please, to find the least of those which have the same ratio with them.

344 Let A , B , C be the given numbers, as many as we please; thus it is required to find the least of those which have the same ratio with A , B , C .

346 Now, if A , B , C are prime to one another, they are the least of those which have the same ratio with them. [ VII. 21 ]

347 But, if not, let D the greatest common measure of A , B , C be taken, [ VII. 3 ] and, as many times as D measures the numbers A , B , C respectively, so many units let there be in the numbers E , F , G respectively.

348 Therefore the numbers E , F , G measure the numbers A , B , C respectively according to the units in D . [ VII. 16 ]

349 Therefore E , F , G measure A , B , C the same number of times; therefore E , F , G are in the same ratio with A , B , C . [ VII. Def. 20 ]

350 I say next that they are the least that are in that ratio.

351 For, if E , F , G are not the least of those which have the same ratio with A , B , C , there will be numbers less than E , F , G which are in the same ratio with A , B , C .

352 Let them be H , K , L ; therefore H measures A the same number of times that the numbers K , L measure the numbers B , C respectively.

353 Now, as many times as H measures A , so many units let there be in M ; therefore the numbers K , L also measure the numbers B , C respectively according to the units in M .

354 And, since H measures A according to the units in M , therefore M also measures A according to the units in H . [ VII. 16 ]

355 For the same reason M also measures the numbers B , C according to the units in the numbers K , L respectively;

357 Now, since H measures A according to the units in M , therefore H by multiplying M has made A . [ VII. Def. 15 ]

358 For the same reason also E by multiplying D has made A .

359 Therefore the product of E , D is equal to the product of H , M .

360 Therefore, as E is to H , so is M to D . [ VII. 19 ]

361 But E is greater than H ; therefore M is also greater than D .

362 And it measures A , B , C : which is impossible, for by hypothesis D is the greatest common measure of A , B , C .

363 Therefore there cannot be any numbers less than E , F , G which are in the same ratio with A , B , C .

364 Therefore E , F , G are the least of those which have the same ratio with A , B , C . Q. E. D.

PROPOSITION 34.

365 Given two numbers, to find the least number which they measure.

366 Let A , B be the two given numbers; thus it is required to find the least number which they measure.

368 First, let A , B be prime to one another, and let A by multiplying B make C ; therefore also B by multiplying A has made C . [ VII. 16 ]

370 I say next that it is also the least number they measure.

371 For, if not, A , B will measure some number which is less than C .

373 Then, as many times as A measures D , so many units let there be in E , and, as many times as B measures D , so many units let there be in F ; therefore A by multiplying E has made D , and B by multiplying F has made D ; [ VII. Def. 15 ] therefore the product of A , E is equal to the product of B , F .

375 But A , B are prime, primes are also least, [ VII. 21 ] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [ VII. 20 ] therefore B measures E , as consequent consequent.

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

377 But B measures E ; therefore C also measures D , the greater the less: which is impossible.

378 Therefore A , B do not measure any number less than C ; therefore C is the least that is measured by A , B .

379 Next, let A , B not be prime to one another, and let F , E , the least numbers of those which have the same ratio with A , B , be taken; [ VII. 33 ] therefore the product of A , E is equal to the product of B , F . [ VII. 19 ]

380 And let A by multiplying E make C ; therefore also B by multiplying F has made C ; therefore A , B measure C .

381 I say next that it is also the least number that they measure.

382 For, if not, A , B will measure some number which is less than C .

384 And, as many times as A measures D , so many units let there be in G , and, as many times as B measures D , so many units let there be in H .

385 Therefore A by multiplying G has made D , and B by multiplying H has made D .

386 Therefore the product of A , G is equal to the product of B , H ; therefore, as A is to B , so is H to G . [ VII. 19 ]

389 But F , E are least, and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [ VII. 20 ] therefore E measures G .

390 And, since A by multiplying E , G has made C , D , therefore, as E is to G , so is C to D . [ VII. 17 ]

391 But E measures G ; therefore C also measures D , the greater the less: which is impossible.

392 Therefore A , B will not measure any number which is less than C .

393 Therefore C is the least that is measured by A , B . Q. E. D.

PROPOSITION 35.

394 If two numbers measure any number, the least number measured by them will also measure the same.

395 For let the two numbers A , B measure any number CD , and let E be the least that they measure; I say that E also measures CD .

396 For, if E does not measure CD , let E , measuring DF , leave CF less than itself.

397 Now, since A , B measure E , and E measures DF , therefore A , B will also measure DF .