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1 If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the numbers are the least of those which have the same ratio with them.

2 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 A , B , C , D are the least of those which have the same ratio with them.

3 For, if not, let E , F , G , H be less than A , B , C , D , and in the same ratio with them.

4 Now, since A , B , C , D are in the same ratio with E , F , G , H , and the multitude of the numbers A , B , C , D is equal to the multitude of the numbers E , F , G , H , therefore, ex aequali , as A is to D , so is E to H . [ VII. 14 ]

5 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 greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent. [ VII. 20 ]

6 Therefore A measures E , the greater the less: which is impossible.

7 Therefore E , F , G , H which are less than A , B , C , D are not in the same ratio with them.

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

9 To find numbers in continued proportion, as many as may be prescribed, and the least that are in a given ratio.

10 Let the ratio of A to B be the given ratio in least numbers; thus it is required to find numbers in continued proportion, as many as may be prescribed, and the least that are in the ratio of A to B .

11 Let four be prescribed; let A by multiplying itself make C , and by multiplying B let it make D ; let B by multiplying itself make E ; further, let A by multiplying C , D , E make F , G , H , and let B by multiplying E make K .

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

13 Again, since A by multiplying B has made D , and B by multiplying itself has made E , therefore the numbers A , B by multiplying B have made the numbers D , E respectively.

14 Therefore, as A is to B , so is D to E . [ VII. 18 ]

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

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

17 But, as C is to D , so was A to B ; therefore also, as A is to B , so is F to G .

18 Again, since A by multiplying D , E has made G , H , therefore, as D is to E , so is G to H . [ VII. 17 ]

19 But, as D is to E , so is A to B .

20 Therefore also, as A is to B , so is G to H .

21 And, since A , B by multiplying E have made H , K , therefore, as A is to B , so is H to K . [ VII. 18 ]

22 But, as A is to B , so is F to G , and G to H .

23 Therefore also, as F is to G , so is G to H , and H to K ; therefore C , D , E , and F , G , H , K are proportional in the ratio of A to B .

24 I say next that they are the least numbers that are so.

25 For, since A , B are the least of those which have the same ratio with them, and the least of those which have the same ratio are prime to one another, [ VII. 22 ] therefore A , B are prime to one another.

26 And the numbers A , B by multiplying themselves respectively have made the numbers C , E , and by multiplying the numbers C , E respectively have made the numbers F , K ; therefore C , E and F , K are prime to one another respectively. [ VII. 27 ]

27 But, if there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, they are the least of those which have the same ratio with them. [ VIII. 1 ]

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

30 If as many numbers as we please in continued proportion be the least of those which have the same ratio with them, the extremes of them are prime to one another .

31 Let as many numbers as we please, A , B , C , D , in continued proportion be the least of those which have the same ratio with them; I say that the extremes of them A , D are prime to one another.

32 For let two numbers E , F , the least that are in the ratio of A , B , C , D , be taken, [ VII. 33 ] then three others G , H , K with the same property; and others, more by one continually, [ VIII. 2 ] until the multitude taken becomes equal to the multitude of the numbers A , B , C , D .

33 Let them be taken, and let them be L , M , N , O .

34 Now, since E , F are the least of those which have the same ratio with them, they are prime to one another. [ VII. 22 ]

35 And, since the numbers E , F by multiplying themselves respectively have made the numbers G , K , and by multiplying the numbers G , K respectively have made the numbers L , O , [ VIII. 2, Por. ] therefore both G , K and L , O are prime to one another. [ VII. 27 ]

36 And, since A , B , C , D are the least of those which have the same ratio with them, while L , M , N , O are the least that are in the same ratio with A , B , C , D , and the multitude of the numbers A , B , C , D is equal to the multitude of the numbers L , M , N , O , therefore the numbers A , B , C , D are equal to the numbers L , M , N , O respectively; therefore A is equal to L , and D to O .

37 And L , O are prime to one another.

38 Therefore A , D are also prime to one another. Q. E. D.

39 Given as many ratios as we please in least numbers, to find numbers in continued proportion which are the least in the given ratios.

40 Let the given ratios in least numbers be that of A to B , that of C to D , and that of E to F ; thus it is required to find numbers in continued proportion which are the least that are in the ratio of A to B , in the ratio of C to D , and in the ratio of E to F .

41 Let G , the least number measured by B , C , be taken. [ VII. 34 ]

42 And, as many times as B measures G , so many times also let A measure H , and, as many times as C measures G , so many times also let D measure K .

43 Now E either measures or does not measure K .

44First, let it measure it.

45 And, as many times as E measures K , so many times let F measure L also.

46 Now, since A measures H the same number of times that B measures G , therefore, as A is to B , so is H to G . [ VII. Def. 20, VII. 13 ]

47 For the same reason also, as C is to D , so is G to K , and further, as E is to F , so is K to L ; therefore H , G , K , L are continuously proportional in the ratio of A to B , in the ratio of C to D , and in the ratio of E to F .

48 I say next that they are also the least that have this property.

49 For, if H , G , K , L are not the least numbers continuously proportional in the ratios of A to B , of C to D , and of E to F , let them be N , O , M , P .

50 Then since, as A is to B , so is N to O , while A , B are least, and the least numbers 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; therefore B measures O . [ VII. 20 ]

51 For the same reason C also measures O ; therefore B , C measure O ; therefore the least number measured by B , C will also measure O . [ VII. 35 ]

52 But G is the least number measured by B , C ; therefore G measures O , the greater the less: which is impossible.

53 Therefore there will be no numbers less than H , G , K , L which are continuously in the ratio of A to B , of C to D , and of E to F .

54 Next, let E not measure K .

55 Let M , the least number measured by E , K , be taken.

56 And, as many times as K measures M , so many times let H , G measure N , O respectively, and, as many times as E measures M , so many times let F measure P also.

57 Since H measures N the same number of times that G measures O , therefore, as H is to G , so is N to O . [ VII. 13 and Def. 20 ]

58 But, as H is to G , so is A to B ; therefore also, as A is to B , so is N to O .

59 For the same reason also, as C is to D , so is O to M .

60 Again, since E measures M the same number of times that F measures P , therefore, as E is to F , so is M to P ; [ VII. 13 and Def. 20 ] therefore N , O , M , P are continuously proportional in the ratios of A to B , of C to D , and of E to F .

61 I say next that they are also the least that are in the ratios A : B , C : D , E : F .

62 For, if not, there will be some numbers less than N , O , M , P continuously proportional in the ratios A : B , C : D , E : F .

63 Let them be Q , R , S , T .

64 Now since, as Q is to R , so is A to B , while A , B are least, 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 B measures R .

65 For the same reason C also measures R ; therefore B , C measure R .

66 Therefore the least number measured by B , C will also measure R . [ VII. 35 ]

67 But G is the least number measured by B , C ; therefore G measures R .

68 And, as G is to R , so is K to S : [ VII. 13 ] therefore K also measures S .

69 But E also measures S ; therefore E , K measure S .

70 Therefore the least number measured by E , K will also measure S . [ VII. 35 ]

71 But M is the least number measured by E , K ; therefore M measures S , the greater the less: which is impossible.

72 Therefore there will not be any numbers less than N , O , M , P continuously proportional in the ratios of A to B , of C to D , and of E to F ; therefore N , O , M , P are the least numbers continuously proportional in the ratios A : B , C : D , E : F . Q. E. D.

73 Plane numbers have to one another the ratio compounded of the ratios of their sides.

74 Let A , B be plane numbers, and let the numbers C , D be the sides of A , and E , F of B ; I say that A has to B the ratio compounded of the ratios of the sides.

75 For, the ratios being given which C has to E and D to F , let the least numbers G , H , K that are continuously in the ratios C : E , D : F be taken, so that, as C is to E , so is G to H , and, as D is to F , so is H to K . [ VIII. 4 ]

76 And let D by multiplying E make L .

77 Now, since D by multiplying C has made A , and by multiplying E has made L , therefore, as C is to E , so is A to L . [ VII. 17 ]

78 But, as C is to E , so is G to H ; therefore also, as G is to H , so is A to L .

79 Again, since E by multiplying D has made L , and further by multiplying F has made B , therefore, as D is to F , so is L to B . [ VII. 17 ]

80 But, as D is to F , so is H to K ; therefore also, as H is to K , so is L to B .

81 But it was also proved that, as G is to H , so is A to L ; therefore, ex aequali , as G is to K , so is A to B . [ VII. 14 ]

82 But G has to K the ratio compounded of the ratios of the sides; therefore A also has to B the ratio compounded of the ratios of the sides. Q. E. D.

83 If there be as many numbers as we please in continued proportion, and the first do not measure the second, neither will any other measure any other.

84 Let there be as many numbers as we please, A , B , C , D , E , in continued proportion, and let A not measure B ; I say that neither will any other measure any other.

85 Now it is manifest that A , B , C , D , E do not measure one another in order; for A does not even measure B .

86 I say, then, that neither will any other measure any other.

87 For, if possible, let A measure C .

88 And, however many A , B , C are, let as many numbers F , G , H , the least of those which have the same ratio with A , B , C , be taken. [ VII. 33 ]

89 Now, since F , G , H are in the same ratio with A , B , C , and the multitude of the numbers A , B , C is equal to the multitude of the numbers F , G , H , therefore, ex aequali , as A is to C , so is F to H . [ VII. 14 ]

90 And since, as A is to B , so is F to G , while A does not measure B , therefore neither does F measure G ; [ VII. Def. 20 ] therefore F is not an unit, for the unit measures any number.

91 Now F , H are prime to one another. [ VIII. 3 ]

92 And, as F is to H , so is A to C ; therefore neither does A measure C .

93 Similarly we can prove that neither will any other measure any other. Q. E. D.

94 If there be as many numbers as we please in continued proportion, and the first measure the last, it will measure the second also.

95 Let there be as many numbers as we please, A , B , C , D , in continued proportion; and let A measure D ; I say that A also measures B .

96 For, if A does not measure B , neither will any other of the numbers measure any other. [ VIII. 6 ]

97 But A measures D .

98 Therefore A also measures B . Q. E. D.

99 If between two numbers there fall numbers in continued proportion with them, then, however many numbers fall between them in continued proportion, so many will also fall in continued proportion between the numbers which have the same ratio with the original numbers.

100 Let the numbers C , D fall between the two numbers A , B in continued proportion with them, and let E be made in the same ratio to F as A is to B ; I say that, as many numbers as have fallen between A , B in continued proportion, so many will also fall between E , F in continued proportion.

101 For, as many as A , B , C , D are in multitude, let so many numbers G , H , K , L , the least of those which have the same ratio with A , C , D , B , be taken; [ VII. 33 ] therefore the extremes of them G , L are prime to one another. [ VIII. 3 ]

102 Now, since A , C , D , B are in the same ratio with G , H , K , L , and the multitude of the numbers A , C , D , B is equal to the multitude of the numbers G , H , K , L , therefore, ex aequali , as A is to B , so is G to L . [ VII. 14 ]

103 But, as A is to B , so is E to F ; therefore also, as G is to L , so is E to F .

104 But G , L 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 greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent. [ VII. 20 ]

105 Therefore G measures E the same number of times as L measures F .

106 Next, as many times as G measures E , so many times let H , K also measure M , N respectively; therefore G , H , K , L measure E , M , N , F the same number of times.

107 Therefore G , H , K , L are in the same ratio with E , M , N , F . [ VII. Def. 20 ]

108 But G , H , K , L are in the same ratio with A , C , D , B ; therefore A , C , D , B are also in the same ratio with E , M , N , F .

109 But A , C , D , B are in continued proportion; therefore E , M , N , F are also in continued proportion.

110 Therefore, as many numbers as have fallen between A , B in continued proportion with them, so many numbers have also fallen between E , F in continued proportion. Q. E. D.

111 If two numbers be prime to one another, and numbers fall between them in continued proportion, then, however many numbers fall between them in continued proportion, so many will also fall between each of them and an unit in continued proportion.

112 Let A , B be two numbers prime to one another, and let C , D fall between them in continued proportion, and let the unit E be set out; I say that, as many numbers as fall between A , B in continued proportion, so many will also fall between either of the numbers A , B and the unit in continued proportion.

113 For let two numbers F , G , the least that are in the ratio of A , C , D , B , be taken, three numbers H , K , L with the same property, and others more by one continually, until their multitude is equal to the multitude of A , C , D , B . [ VIII. 2 ]

114 Let them be taken, and let them be M , N , O , P .

115 It is now manifest that F by multiplying itself has made H and by multiplying H has made M , while G by multiplying itself has made L and by multiplying L has made P . [ VIII. 2, Por. ]

116 And, since M , N , O , P are the least of those which have the same ratio with F , G , and A , C , D , B are also the least of those which have the same ratio with F , G , [ VIII. 1 ] while the multitude of the numbers M , N , O , P is equal to the multitude of the numbers A , C , D , B , therefore M , N , O , P are equal to A , C , D , B respectively; therefore M is equal to A , and P to B .

117 Now, since F by multiplying itself has made H , therefore F measures H according to the units in F .

118 But the unit E also measures F according to the units in it; therefore the unit E measures the number F the same number of times as F measures H .

119 Therefore, as the unit E is to the number F , so is F to H . [ VII. Def. 20 ]

120 Again, since F by multiplying H has made M , therefore H measures M according to the units in F .

121 But the unit E also measures the number F according to the units in it; therefore the unit E measures the number F the same number of times as H measures M .

122 Therefore, as the unit E is to the number F , so is H to M .

123 But it was also proved that, as the unit E is to the number F , so is F to H ; therefore also, as the unit E is to the number F , so is F to H , and H to M .

124 But M is equal to A ; therefore, as the unit E is to the number F , so is F to H , and H to A .

125 For the same reason also, as the unit E is to the number G , so is G to L and L to B .

126 Therefore, as many numbers as have fallen between A , B in continued proportion, so many numbers also have fallen between each of the numbers A , B and the unit E in continued proportion. Q. E. D.

127 If numbers fall between each of two numbers and an unit in continued proportion, however many numbers fall between each of them and an unit in continued proportion, so many also will fall between the numbers themselves in continued proportion.

128 For let the numbers D , E and F , G respectively fall between the two numbers A , B and the unit C in continued proportion; I say that, as many numbers as have fallen between each of the numbers A , B and the unit C in continued proportion, so many numbers will also fall between A , B in continued proportion.

129 For let D by multiplying F make H , and let the numbers D , F by multiplying H make K , L respectively.

130 Now, since, as the unit C is to the number D , so is D to E , therefore the unit C measures the number D the same number of times as D measures E . [ VII. Def. 20 ]

131 But the unit C measures the number D according to the units in D ; therefore the number D also measures E according to the units in D ; therefore D by multiplying itself has made E .

132 Again, since, as C is to the number D , so is E to A , therefore the unit C measures the number D the same number of times as E measures A .

133 But the unit C measures the number D according to the units in D ; therefore E also measures A according to the units in D ; therefore D by multiplying E has made A .

134 For the same reason also F by multiplying itself has made G , and by multiplying G has made B .

135 And, since D by multiplying itself has made E and by multiplying F has made H , therefore, as D is to F , so is E to H . [ VII. 17 ]

136 For the same reason also, as D is to F , so is H to G . [ VII. 18 ]

137 Therefore also, as E is to H , so is H to G .

138 Again, since D by multiplying the numbers E , H has made A , K respectively, therefore, as E is to H , so is A to K . [ VII. 17 ]

139 But, as E is to H , so is D to F ; therefore also, as D is to F , so is A to K .

140 Again, since the numbers D , F by multiplying H have made K , L respectively, therefore, as D is to F , so is K to L . [ VII. 18 ]

141 But, as D is to F , so is A to K ; therefore also, as A is to K , so is K to L .

142 Further, since F by multiplying the numbers H , G has made L , B respectively, therefore, as H is to G , so is L to B . [ VII. 17 ]

143 But, as H is to G , so is D to F ; therefore also, as D is to F , so is L to B .

144 But it was also proved that, as D is to F , so is A to K and K to L ; therefore also, as A is to K , so is K to L and L to B .

145 Therefore A , K , L , B are in continued proportion.

146 Therefore, as many numbers as fall between each of the numbers A , B and the unit C in continued proportion, so many also will fall between A , B in continued proportion. Q. E. D.

147 Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate of that which the side has to the side.

148 Let A , B be square numbers, and let C be the side of A , and D of B ; I say that between A , B there is one mean proportional number, and A has to B the ratio duplicate of that which C has to D .

149 For let C by multiplying D make E .

150 Now, since A is a square and C is its side, therefore C by multiplying itself has made A .

151 For the same reason also D by multiplying itself has made B .

152 Since then C by multiplying the numbers C , D has made A , E respectively, therefore, as C is to D , so is A to E . [ VII. 17 ]

153 For the same reason also, as C is to D , so is E to B . [ VII. 18 ]

154 Therefore also, as A is to E , so is E to B .

155 Therefore between A , B there is one mean proportional number.

156 I say next that A also has to B the ratio duplicate of that which C has to D .

157 For, since A , E , B are three numbers in proportion, therefore A has to B the ratio duplicate of that which A has to E . [ V. Def. 9 ]

158 But, as A is to E , so is C to D .

159 Therefore A has to B the ratio duplicate of that which the side C has to D . Q. E. D.

160 Between two cube numbers there are two mean proportional numbers, and the cube has to the cube the ratio triplicate of that which the side has to the side.

161 Let A , B be cube numbers, and let C be the side of A , and D of B ; I say that between A , B there are two mean proportional numbers, and A has to B the ratio triplicate of that which C has to D .

162 For let C by multiplying itself make E , and by multiplying D let it make F ; let D by multiplying itself make G , and let the numbers C , D by multiplying F make H , K respectively.

163 Now, since A is a cube, and C its side, and C by multiplying itself has made E , therefore C by multiplying itself has made E and by multiplying E has made A .

164 For the same reason also D by multiplying itself has made G and by multiplying G has made B .

165 And, since C by multiplying the numbers C , D has made E , F respectively, therefore, as C is to D , so is E to F . [ VII. 17 ]

166 For the same reason also, as C is to D , so is F to G . [ VII. 18 ]

167 Again, since C by multiplying the numbers E , F has made A , H respectively, therefore, as E is to F , so is A to H . [ VII. 17 ]

168 But, as E is to F , so is C to D .

169 Therefore also, as C is to D , so is A to H .

170 Again, since the numbers C , D by multiplying F have made H , K respectively, therefore, as C is to D , so is H to K . [ VII. 18 ]

171 Again, since D by multiplying each of the numbers F , G has made K , B respectively, therefore, as F is to G , so is K to B . [ VII. 17 ]

172 But, as F is to G , so is C to D ; therefore also, as C is to D , so is A to H , H to K , and K to B .

173 Therefore H , K are two mean proportionals between A , B .

174 I say next that A also has to B the ratio triplicate of that which C has to D .

175 For, since A , H , K , B are four numbers in proportion, therefore A has to B the ratio triplicate of that which A has to H . [ V. Def. 10 ]

176 But, as A is to H , so is C to D ; therefore A also has to B the ratio triplicate of that which C has to D . Q. E. D.

177 If there be as many numbers as we please in continued proportion, and each by multiplying itself make some number, the products will be proportional; and, if the original numbers by multiplying the products make certain numbers, the latter will also be proportional.

178 Let there be as many numbers as we please, A , B , C , in continued proportion, so that, as A is to B , so is B to C ; let A , B , C by multiplying themselves make D , E , F , and by multiplying D , E , F let them make G , H , K ; I say that D , E , F and G , H , K are in continued proportion.

179 For let A by multiplying B make L , and let the numbers A , B by multiplying L make M . N respectively.

180 And again let B by multiplying C make O , and let the numbers B , C by multiplying O make P , Q respectively.

181 Then, in manner similar to the foregoing, we can prove that D , L , E and G , M , N , H are continuously proportional in the ratio of A to B , and further E , O , F and H , P , Q , K are continuously proportional in the ratio of B to C .

182 Now, as A is to B , so is B to C ; therefore D , L , E are also in the same ratio with E , O , F , and further G , M , N , H in the same ratio with H , P , Q , K .

183 And the multitude of D , L , E is equal to the multitude of E , O , F , and that of G , M , N , H to that of H , P , Q , K ; therefore, ex acquali , as D is to E , so is E to F , and, as G is to H , so is H to K . [ VII. 14 ] Q. E. D.

184 If a square measure a square, the side will also measure the side; and, if the side measure the side, the square will also measure the square.

185 Let A , B be square numbers, let C , D be their sides, and let A measure B ; I say that C also measures D .

186 For let C by multiplying D make E ; therefore A , E , B are continuously proportional in the ratio of C to D . [ VIII. 11 ]

187 And, since A , E , B are continuously proportional, and A measures B , therefore A also measures E . [ VIII. 7 ]

188 And, as A is to E , so is C to D ; therefore also C measures D . [ VII. Def. 20 ]

189 Again, let C measure D ; I say that A also measures B .

190 For, with the same construction, we can in a similar manner prove that A , E , B are continuously proportional in the ratio of C to D .

191 And since, as C is to D , so is A to E , and C measures D , therefore A also measures E . [ VII. Def. 20 ]

192 And A , E , B are continuously proportional; therefore A also measures B .

193Therefore etc. Q. E. D.

194 If a cube number measure a cube number, the side will also measure the side; and, if the side measure the side, the cube will also measure the cube.

195 For let the cube number A measure the cube B , and let C be the side of A and D of B ; I say that C measures D .

196 For let C by multiplying itself make E , and let D by multiplying itself make G ; further, let C by multiplying D make F , and let C , D by multiplying F make H , K respectively.

197 Now it is manifest that E , F , G and A , H , K , B are continuously proportional in the ratio of C to D . [ VIII. 11, 12 ]

198 And, since A , H , K , B are continuously proportional, and A measures B , therefore it also measures H . [ VIII. 7 ]

199 And, as A is to H , so is C to D ; therefore C also measures D . [ VII. Def. 20 ]

200 Next, let C measure D ; I say that A will also measure B .

201 For, with the same construction, we can prove in a similar manner that A , H , K , B are continuously proportional in the ratio of C to D .

202 And, since C measures D , and, as C is to D , so is A to H , therefore A also measures H , [ VII. Def. 20 ] so that A measures B also. Q. E. D.

203 If a square number do not measure a square number, neither will the side measure the side; and, if the side do not measure the side, neither will the square measure the square.

204 Let A , B be square numbers, and let C , D be their sides; and let A not measure B ; I say that neither does C measure D .

205 For, if C measures D , A will also measure B . [ VIII. 14 ]

206 But A does not measure B ; therefore neither will C measure D .

207 Again, let C not measure D ; I say that neither will A measure B .

208 For, if A measures B , C will also measure D . [ VIII. 14 ]

209 But C does not measure D ; therefore neither will A measure B . Q. E. D.

210 If a cube number do not measure a cube number, neither will the side measure the side; and, if the side do not measure the side, neither will the cube measure the cube.

211 For let the cube number A not measure the cube number B , and let C be the side of A , and D of B ; I say that C will not measure D .

212 For if C measures D , A will also measure B . [ VIII. 15 ]

213 But A does not measure B ; therefore neither does C measure D .

214 Again, let C not measure D ; I say that neither will A measure B .

215 For, if A measures B , C will also measure D . [ VIII. 15 ]

216 But C does not measure D ; therefore neither will A measure B . Q. E. D.

217 Between two similar plane numbers there is one mean proportional number; and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side.

218 Let A , B be two similar plane numbers, and let the numbers C , D be the sides of A , and E , F of B .

219 Now, since similar plane numbers are those which have their sides proportional, [ VII. Def. 21 ] therefore, as C is to D , so is E to F .

220 I say then that between A , B there is one mean proportional number, and A has to B the ratio duplicate of that which C has to E , or D to F , that is, of that which the corresponding side has to the corresponding side.

221 Now since, as C is to D , so is E to F , therefore, alternately, as C is to E , so is D to F . [ VII. 13 ]

222 And, since A is plane, and C , D are its sides, therefore D by multiplying C has made A .

223 For the same reason also E by multiplying F has made B .

224 Now let D by multiplying E make G .

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

226 But, as C is to E , so is D to F ; therefore also, as D is to F , so is A to G .

227 Again, since E by multiplying D has made G , and by multiplying F has made B , therefore, as D is to F , so is G to B . [ VII. 17 ]

228 But it was also proved that, as D is to F , so is A to G ; therefore also, as A is to G , so is G to B .

229 Therefore A , G , B are in continued proportion.

230 Therefore between A , B there is one mean proportional number.

231 I say next that A also has to B the ratio duplicate of that which the corresponding side has to the corresponding side, that is, of that which C has to E or D to F .

232 For, since A , G , B are in continued proportion, A has to B the ratio duplicate of that which it has to G . [ V. Def. 9 ]

233 And, as A is to G , so is C to E , and so is D to F .

234 Therefore A also has to B the ratio duplicate of that which C has to E or D to F . Q. E. D.

235 Between two similar solid numbers there fall two mean proportional numbers; and the solid number has to the similar solid number the ratio triplicate of that which the corresponding side has to the corresponding side.

236 Let A , B be two similar solid numbers, and let C , D , E be the sides of A , and F , G , H of B .

237 Now, since similar solid numbers are those which have their sides proportional, [ VII. Def. 21 ] therefore, as C is to D , so is F to G , and, as D is to E , so is G to H .

238 I say that between A , B there fall two mean proportional numbers, and A has to B the ratio triplicate of that which C has to F , D to G , and also E to H .

239 For let C by multiplying D make K , and let F by multiplying G make L .

240 Now, since C , D are in the same ratio with F , G , and K is the product of C , D , and L the product of F , G , K , L are similar plane numbers; [ VII. Def. 21 ] therefore between K , L there is one mean proportional number. [ VIII. 18 ]

241 Let it be M

242 Therefore M is the product of D , F , as was proved in the theorem preceding this. [ VIII. 18 ]

243 Now, since D by multiplying C has made K , and by multiplying F has made M , therefore, as C is to F , so is K to M . [ VII. 17 ]

244 But, as K is to M , so is M to L .

245 Therefore K , M , L are continuously proportional in the ratio of C to F .

246 And since, as C is to D , so is F to G , alternately therefore, as C is to F , so is D to G . [ VII. 13 ]

247 For the same reason also, as D is to G , so is E to H .

248 Therefore K , M , L are continuously proportional in the ratio of C to F , in the ratio of D to G , and also in the ratio of E to H .

249 Next, let E , H by multiplying M make N , O respectively.

250 Now, since A is a solid number, and C , D , E are its sides, therefore E by multiplying the product of C , D has made A .

251 But the product of C , D is K ; therefore E by multiplying K has made A .

252 For the same reason also H by multiplying L has made B .

253 Now, since E by multiplying K has made A , and further also by multiplying M has made N , therefore, as K is to M , so is A to N . [ VII. 17 ]

254 But, as K is to M , so is C to F , D to G , and also E to H ; therefore also, as C is to F , D to G , and E to H , so is A to N .

255 Again, since E , H by multiplying M have made N , O respectively, therefore, as E is to H , so is N to O . [ VII. 18 ]

256 But, as E is to H , so is C to F and D to G ; therefore also, as C is to F , D to G , and E to H , so is A to N and N to O .

257 Again, since H by multiplying M has made O , and further also by multiplying L has made B , therefore, as M is to L , so is O to B . [ VII. 17 ]

258 But, as M is to L , so is C to F , D to G , and E to H .

259 Therefore also, as C is to F , D to G , and E to H , so not only is O to B , but also A to N and N to O .

260 Therefore A , N , O , B are continuously proportional in the aforesaid ratios of the sides.

261 I say that A also has to B the ratio triplicate of that which the corresponding side has to the corresponding side, that is, of the ratio which the number C has to F , or D to G , and also E to H .

262 For, since A , N , O , B are four numbers in continued proportion, therefore A has to B the ratio triplicate of that which A has to N . [ V. Def. 10 ]

263 But, as A is to N , so it was proved that C is to F , D to G , and also E to H .

264 Therefore A also has to B the ratio triplicate of that which the corresponding side has to the corresponding side, that is, of the ratio which the number C has to F , D to G , and also E to H . Q. E. D.

265 If one mean proportional number fall between two numbers, the numbers will be similar plane numbers.

266 For let one mean proportional number C fall between the two numbers A , B ; I say that A , B are similar plane numbers.

267 Let D , E , the least numbers of those which have the same ratio with A , C , be taken; [ VII. 33 ] therefore D measures A the same number of times that E measures C . [ VII. 20 ]

268 Now, as many times as D measures A , so many units let there be in F ; therefore F by multiplying D has made A , so that A is plane, and D , F are its sides.

269 Again, since D , E are the least of the numbers which have the same ratio with C , B , therefore D measures C the same number of times that E measures B . [ VII. 20 ]

270 As many times, then, as E measures B , so many units let there be in G ; therefore E measures B according to the units in G ; therefore G by multiplying E has made B .

271 Therefore B is plane, and E , G are its sides.

272 Therefore A , B are plane numbers.

273 I say next that they are also similar.

274 For, since F by multiplying D has made A , and by multiplying E has made C , therefore, as D is to E , so is A to C , that is, C to B . [ VII. 17 ]

275 Again, since E by multiplying F , G has made C , B respectively, therefore, as F is to G , so is C to B . [ VII. 17 ]

276 But, as C is to B , so is D to E ; therefore also, as D is to E , so is F to G .

277 And alternately, as D is to F , so is E to G . [ VII. 13 ]

278 Therefore A , B are similar plane numbers; for their sides are proportional. Q. E. D.

279 If two mean proportional numbers fall between two numbers, the numbers are similar solid numbers.

280 For let two mean proportional numbers C , D fall between the two numbers A , B ; I say that A , B are similar solid numbers.

281 For let three numbers E , F , G , the least of those which have the same ratio with A , C , D , be taken; [ VII. 33 or VIII. 2 ] therefore the extremes of them E , G are prime to one another. [ VIII. 3 ]

282 Now, since one mean proportional number F has fallen between E , G , therefore E , G are similar plane numbers. [ VIII. 20 ]

283 Let, then, H , K be the sides of E , and L , M of G .

284 Therefore it is manifest from the theorem before this that E , F , G are continuously proportional in the ratio of H to L and that of K to M .

285 Now, since E , F , G are the least of the numbers which have the same ratio with A , C , D , and the multitude of the numbers E , F , G is equal to the multitude of the numbers A , C , D , therefore, ex aequali , as E is to G , so is A to D . [ VII. 14 ]

286 But E , G are prime, primes are also least, [ VII. 21 ] and the least measure those which have the same ratio with them 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 A the same number of times that G measures D .

287 Now, as many times as E measures A , so many units let there be in N .

288 Therefore N by multiplying E has made A .

289 But E is the product of H , K ; therefore N by multiplying the product of H , K has made A .

290 Therefore A is solid, and H , K , N are its sides.

291 Again, since E , F , G are the least of the numbers which have the same ratio as C , D , B , therefore E measures C the same number of times that G measures B .

292 Now, as many times as E measures C , so many units let there be in O .

293 Therefore G measures B according to the units in O ; therefore O by multiplying G has made B .

294 But G is the product of L , M ; therefore O by multiplying the product of L , M has made B .

295 Therefore B is solid, and L , M , O are its sides; therefore A , B are solid.

296I say that they are also similar.

297 For since N , O by multiplying E have made A , C , therefore, as N is to O , so is A to C , that is, E to F . [ VII. 18 ]

298 But, as E is to F , so is H to L and K to M ; therefore also, as H is to L , so is K to M and N to O .

299 And H , K , N are the sides of A , and O , L , M the sides of B .

300 Therefore A , B are similar solid numbers. Q. E. D.

301 If three numbers be in continued proportion, and the first be square, the third will also be square.

302 Let A , B , C be three numbers in continued proportion, and let A the first be square; I say that C the third is also square.

303 For, since between A , C there is one mean proportional number, B , therefore A , C are similar plane numbers. [ VIII. 20 ]

304 But A is square; therefore C is also square. Q. E. D.

305 If four numbers be in continued proportion, and the first be cube, the fourth will also be cube.

306 Let A , B , C , D be four numbers in continued proportion, and let A be cube; I say that D is also cube.

307 For, since between A , D there are two mean proportional numbers B , C , therefore A , D are similar solid numbers. [ VIII. 21 ]

308 But A is cube; therefore D is also cube. Q. E. D.

309 If two numbers have to one another the ratio which a square number has to a square number, and the first be square, the second will also be square.

310 For let the two numbers A , B have to one another the ratio which the square number C has to the square number D , and let A be square; I say that B is also square.

311 For, since C , D are square, C , D are similar plane numbers.

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

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

314 And A is square; therefore B is also square. [ VIII. 22 ] Q. E. D.

315 If two numbers have to one another the ratio which a cube number has to a cube number, and the first be cube, the second will also be cube.

316 For let the two numbers A , B have to one another the ratio which the cube number C has to the cube number D , and let A be cube; I say that B is also cube.

317 For, since C , D are cube, C , D are similar solid numbers.

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

319 And, as many numbers as fall between C , D in continued proportion, so many will also fall between those which have the same ratio with them; [ VIII. 8 ] so that two mean proportional numbers fall between A , B also.

320 Let E , F so fall.

321 Since, then, the four numbers A , E , F , B are in continued proportion, and A is cube, therefore B is also cube. [ VIII. 23 ] Q. E. D.

322 Similar plane numbers have to one another the ratio which a square number has to a square number.

323 Let A , B be similar plane numbers; I say that A has to B the ratio which a square number has to a square number.

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

325 Let it so fall, and let it be C ; and let D , E , F , the least numbers of those which have the same ratio with A , C , B , be taken; [ VII. 33 or VIII. 2 ] therefore the extremes of them D , F are square. [ VIII. 2, Por. ]

326 And since, as D is to F , so is A to B , and D , F are square, therefore A has to B the ratio which a square number has to a square number. Q. E. D.

327 Similar solid numbers have to one another the ratio which a cube number has to a cube number.

328 Let A , B be similar solid numbers; I say that A has to B the ratio which a cube number has to a cube number.

329 For, since A , B are similar solid numbers, therefore two mean proportional numbers fall between A , B . [ VIII. 19 ]

330 Let C , D so fall, and let E , F , G , H , the least numbers of those which have the same ratio with A , C , D , B , and equal with them in multitude, be taken; [ VII. 33 or VIII. 2 ] therefore the extremes of them E , H are cube. [ VIII. 2, Por. ]

331 And, as E is to H , so is A to B ; therefore A also has to B the ratio which a cube number has to a cube number. Q. E. D.