G.193 parts ratio multiples — Heath
V.15
Parts have the
same ratio as the same
multiples of them taken in corresponding order.
For let / A B / be the same multiple of C that / D E / is of F ;
I say that, as C is to F , so is / A B / to / D E / .
A
G
H
B
D
K
L
E
C
F
For, since / A B / is the same multiple of C that / D E / is of F , as many magnitudes as there are in / A B / equal to C , so many are there also in / D E / equal to F .
Let / A B / be divided into the magnitudes / A G / , / G H / , / H B / equal to C , and / D E / into the magnitudes / D K / , / K L / , / L E / equal to F ; then the multitude of the magnitudes / A G / , / G H / , / H B / will be equal to the multitude of the magnitudes / D K / , / K L / , / L E / .
And, since / A G / , / G H / , / H B / are equal to one another, and / D K / , / K L / , / L E / are also equal to one another,
[G.183] therefore, as
/ A G / is to
/ D K / , so is
/ G H / to
/ K L / , and
/ H B / to
/ L E / .
Therefore, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents ; [G.190]
therefore, as / A G / is to / D K / , so is / A B / to / D E / .
But / A G / is equal to C and / D K / to F ;
therefore, as C is to F , so is / A B / to / D E / .
Therefore etc. Q. E. D.
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"G.169" [label="G.169", tooltip="G.169 (V.def.1): A magnitude is apart of a magnitude, the less of the greater, when it measures the greater.", fillcolor="#224422", URL="../G.169/index.html", target="_top"];
"G.170" [label="G.170", tooltip="G.170 (V.def.2): The greater is amultiple of the less when it ismeasured by the less.", fillcolor="#224422", URL="../G.170/index.html", target="_top"];
"G.171" [label="G.171", tooltip="G.171 (V.1): If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple one of the magnitudes is of one, ...", fillcolor="#222244", URL="../G.171/index.html", target="_top"];
"G.176" [label="G.176", tooltip="G.176 (V.def.5): Magnitudes are said tobe in the same ratio, the first to the second and the third to the fourth, when, if anyequimultiples whatever be taken of the first and third, and any equimul...", fillcolor="#224422", URL="../G.176/index.html", target="_top"];
"G.180" [label="G.180", tooltip="G.180 (V.def.6): Let magnitudes which have thesame ratio be calledproportional.", fillcolor="#224422", URL="../G.180/index.html", target="_top"];
"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.182" [label="G.182", tooltip="G.182 (V.def.13): Inverse ratio means taking theconsequent as antecedent in relation to the antecedent as consequent.", fillcolor="#224422", URL="../G.182/index.html", target="_top"];
"G.183" [label="G.183", tooltip="G.183 (V.7): Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.", fillcolor="#222244", URL="../G.183/index.html", target="_top"];
"G.190" [label="G.190", tooltip="G.190 (V.12): If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.", fillcolor="#222244", URL="../G.190/index.html", target="_top"];
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"G.171" -> "G.170";
"G.176" -> "G.170";
"G.183" -> "G.170";
"G.190" -> "G.170";
"G.193" -> "G.170";
"G.190" -> "G.171";
"G.180" -> "G.176";
"G.183" -> "G.176";
"G.190" -> "G.176";
"G.193" -> "G.176";
"G.183" -> "G.180";
"G.190" -> "G.180";
"G.182" -> "G.181";
"G.190" -> "G.181";
"G.193" -> "G.181";
"G.183" -> "G.182";
"G.193" -> "G.183";
"G.193" -> "G.190";
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Required for
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