PROPOSITION 15.
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.
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 .
A
B
G
H
C
D
E
K
L
F
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.
Let BC be divided into the units in it, BG , GH , HC , and EF into the numbers EK , KL , LF equal to D .
Thus the multitude of BG , GH , HC will be equal to the multitude of EK , KL , LF .
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 .
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 .
But the unit BG is equal to the unit A , and the number EK to the number D .
Therefore, as the unit A is to the number D , so is BC to EF .
Therefore the unit A measures the number D the same number of times that BC measures EF .
Q. E. D.
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"V.def.2" [label="V.def.2", fillcolor="#224422", tooltip="V.def.2: The greater is a multiple of the less when it is measured by the less.", URL="../../../book_v/definitions/index.html#v-def-2", target="_top"];
"V.def.11" [label="V.def.11", fillcolor="#224422", tooltip="V.def.11: The term corresponding magnitudes is used of antecedents in relation to antecedents, and of consequents in relation to consequents.", URL="../../../book_v/definitions/index.html#v-def-11", target="_top"];
"V.def.12" [label="V.def.12", fillcolor="#224422", tooltip="V.def.12: Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.", URL="../../../book_v/definitions/index.html#v-def-12", target="_top"];
"VII.def.1" [label="VII.def.1", fillcolor="#224422", tooltip="VII.def.1: An unit is that by virtue of which each of the things that exist is called one.", URL="../../definitions/index.html#vii-def-1", target="_top"];
"VII.def.2" [label="VII.def.2", fillcolor="#224422", tooltip="VII.def.2: A number is a multitude composed of units.", URL="../../definitions/index.html#vii-def-2", target="_top"];
"VII.def.3" [label="VII.def.3", fillcolor="#224422", tooltip="VII.def.3: A number is a part of a number, the less of the greater, when it measures the greater.", URL="../../definitions/index.html#vii-def-3", target="_top"];
"VII.def.4" [label="VII.def.4", fillcolor="#224422", tooltip="VII.def.4: but parts when it does not measure it.", URL="../../definitions/index.html#vii-def-4", target="_top"];
"VII.def.5" [label="VII.def.5", fillcolor="#224422", tooltip="VII.def.5: The greater number is a multiple of the less when it is measured by the less.", URL="../../definitions/index.html#vii-def-5", target="_top"];
"VII.def.20" [label="VII.def.20", fillcolor="#224422", tooltip="VII.def.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.", URL="../../definitions/index.html#vii-def-20", target="_top"];
"VII.5" [label="VII.5", fillcolor="#222244", tooltip="VII.5: 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.", URL="../05/index.html", target="_top"];
"VII.6" [label="VII.6", fillcolor="#222244", tooltip="VII.6: 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.", URL="../06/index.html", target="_top"];
"VII.12" [label="VII.12", fillcolor="#222244", tooltip="VII.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.", URL="../12/index.html", target="_top"];
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"V.def.2" -> "V.def.1";
"VII.def.3" -> "V.def.1";
"VII.def.4" -> "V.def.1";
"VII.def.5" -> "V.def.1";
"VII.15" -> "V.def.1";
"VII.5" -> "V.def.2";
"V.def.12" -> "V.def.11";
"VII.12" -> "V.def.11";
"VII.15" -> "V.def.12";
"VII.def.2" -> "VII.def.1";
"VII.15" -> "VII.def.1";
"VII.def.3" -> "VII.def.2";
"VII.def.5" -> "VII.def.2";
"VII.5" -> "VII.def.2";
"VII.6" -> "VII.def.2";
"VII.12" -> "VII.def.2";
"VII.15" -> "VII.def.2";
"VII.def.20" -> "VII.def.3";
"VII.5" -> "VII.def.3";
"VII.def.20" -> "VII.def.4";
"VII.6" -> "VII.def.4";
"VII.def.20" -> "VII.def.5";
"VII.5" -> "VII.def.5";
"VII.12" -> "VII.def.20";
"VII.6" -> "VII.5";
"VII.12" -> "VII.5";
"VII.12" -> "VII.6";
"VII.15" -> "VII.12";
}
Required for
VII.16 VII.18 VII.19 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.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.15
