G.340 many numbers please — Heath

IX.11

If as many numbers as we please beginning from an unit be in continued proportion, the less measures the greater according to some one of the numbers which have place among the proportional numbers.

Let there be as many numbers as we please, B, C, D, E, beginning from the unit A and in continued proportion ;

I say that B, the least of the numbers B, C, D, E, measures E according to some one of the numbers C, D.

A B C D E
For since, as the unit A is to B, so is D to E,
therefore the unit A measures the number B the same number of times as D measures E;
therefore, alternately, the unit A measures D the same number of times as B measures E. [G.268]
But the unit A measures D according to the units in it;
therefore B also measures E according to the units in D;
so that B the less measures E the greater according to some number of those which have place among the proportional numbers.—
PORISM. And it is manifest that, whatever place the measuring number has, reckoned from the unit, the same place also has the number according to which it measures, reckoned from the number measured, in the direction of the number before it.—
Q. E. D.

Dependency Graph

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"G.254" [label="G.254", tooltip="G.254 (VII.def.4): Butparts when it does notmeasure it.", fillcolor="#224422", URL="../G.254/index.html", target="_top"]; "G.256" [label="G.256", tooltip="G.256 (VII.def.5): The greaternumber is amultiple of the less when it ismeasured by the less.", fillcolor="#224422", URL="../G.256/index.html", target="_top"]; "G.257" [label="G.257", tooltip="G.257 (VII.5): If a number be a part of a number, and another be the same part of another, the sum is the same part of the sum that the one is of the one.", fillcolor="#222244", URL="../G.257/index.html", target="_top"]; "G.258" [label="G.258", tooltip="G.258 (VII.6): If a number be parts of a number, and another be the same parts of another, the sum is the same parts of the sum that the one is of the one.", fillcolor="#222244", URL="../G.258/index.html", target="_top"]; "G.263" [label="G.263", tooltip="G.263 (VII.def.20): Numbers areproportional when the first is thesame multiple, or thesame part, or thesame parts, of the second that the third is of the fourth.", fillcolor="#224422", URL="../G.263/index.html", target="_top"]; "G.265" [label="G.265", tooltip="G.265 (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.", fillcolor="#222244", URL="../G.265/index.html", target="_top"]; "G.268" [label="G.268", tooltip="G.268 (VII.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 tim...", fillcolor="#222244", URL="../G.268/index.html", target="_top"]; "G.340" [label="G.340", tooltip="G.340 (IX.11): If as many numbers as we please beginning from an unit be in continued proportion, the less measures the greater according to some one of the numbers which have place among the pro...", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", URL="index.html", target="_top"]; "G.170" -> "G.169"; "G.253" -> "G.169"; "G.254" -> "G.169"; "G.256" -> "G.169"; "G.268" -> "G.169"; 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Required for

G.342, G.343, G.364, G.370