PROPOSITION 12.
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.
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 .
A
B
C
D
For since, as A is to B , so is C to D ,
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 ]
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Required for
VII.15 VII.16 VII.18 VII.19 VII.20 VII.21 VII.24 VII.25 VII.26 VII.27 VII.30 VII.33 VII.34 VII.36 VII.37 VII.38 VII.39 VIII.1 VIII.2 VIII.3 VIII.4 VIII.5 VIII.6 VIII.7 VIII.8 VIII.9 VIII.10 VIII.11 VIII.12 VIII.14 VIII.15 VIII.16 VIII.17 VIII.19 VIII.20 VIII.21 VIII.22 VIII.23 VIII.24 VIII.25 VIII.26 VIII.27 IX.1 IX.2 IX.3 IX.4 IX.5 IX.6 IX.8 IX.9 IX.10 IX.11 IX.12 IX.13 IX.14 IX.15 IX.16 IX.17 IX.18 IX.19 IX.32 IX.35 IX.36 X.9 X.10 X.12 X.13 X.17 X.18 X.22 X.23 X.24 X.25 X.26 X.27 X.28 X.29 X.30 X.31 X.32 X.33 X.34 X.35 X.36 X.37 X.38 X.39 X.40 X.41 X.42 X.43 X.44 X.45 X.46 X.47 X.48 X.49 X.50 X.51 X.52 X.53 X.54 X.55 X.56 X.57 X.58 X.59 X.60 X.61 X.62 X.63 X.64 X.65 X.66 X.67 X.68 X.69 X.70 X.71 X.72 X.73 X.75 X.76 X.77 X.78 X.79 X.80 X.81 X.82 X.83 X.84 X.85 X.86 X.87 X.88 X.89 X.90 X.91 X.92 X.93 X.94 X.95 X.96 X.97 X.98 X.99 X.100 X.101 X.102 X.103 X.104 X.105 X.106 X.107 X.108 X.109 X.110 X.111 X.112 X.113 X.114 XIII.6 XIII.11 XIII.16 XIII.17 XIII.18
See Commentary on Proposition VII.12
