G.261 number part number — Heath
VII.9
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.
For let the number A be a part of the number / B C / , and another, D , the same part of another, / E F / , that A is of / B C / ;
I say that, alternately also, whatever part or parts A is of D , the same part or parts is / B C / of / E F / also.
A
B
G
C
D
E
H
F
For since, whatever part A is of / B C / , the same part also is D of / E F / ,
therefore, as many numbers as there are in / B C / equal to A , so many also are there in / E F / equal to D .
Let / B C / be divided into the numbers equal to A , namely / B G / , / G C / , and / E F / into those equal to D , namely / E H / , / H F / ; thus the multitude of / B G / , / G C / will be equal to the multitude of / E H / , / H F / .
Now, since the numbers / B G / , / G C / are equal to one another, and the numbers / E H / , / H F / are also equal to one another, while the multitude of / B G / , / G C / is equal to the multitude of / E H / , / H F / ,
therefore, whatever part or parts / B G / is of / E H / , the same part or the same parts is / G C / of / H F / also;
so that, in addition, whatever part or parts
/ B G / is of
/ E H / , the same part also, or the same parts, is the sum
/ B C / of the sum
/ E F / .
[G.257 , G.258 ] But / B G / is equal to A , and / E H / to D ;
therefore, whatever part or parts A is of D , the same part or the same parts is / B C / of / E F / also.
Therefore etc. Q. E. D.
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"G.181" [label="G.181", tooltip="G.181 (V.def.11): The termcorresponding magnitudes is used of antecedents in relation to antecedents, and of consequents in relation to consequents.", fillcolor="#224422", URL="../G.181/index.html", target="_top"];
"G.194" [label="G.194", tooltip="G.194 (V.def.12): Alternate ratio means taking theantecedent in relation to the antecedent and the consequent in relation to the consequent.", fillcolor="#224422", URL="../G.194/index.html", target="_top"];
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"G.256" -> "G.169";
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"G.194" -> "G.181";
"G.261" -> "G.194";
"G.248" -> "G.247";
"G.253" -> "G.248";
"G.256" -> "G.248";
"G.257" -> "G.248";
"G.258" -> "G.248";
"G.261" -> "G.248";
"G.254" -> "G.253";
"G.256" -> "G.253";
"G.257" -> "G.253";
"G.258" -> "G.253";
"G.261" -> "G.253";
"G.258" -> "G.254";
"G.261" -> "G.254";
"G.257" -> "G.256";
"G.258" -> "G.257";
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Required for
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