PROPOSITION 1.

If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the numbers are the least of those which have the same ratio with them.

Let there be as many numbers as we please, A, B, C, D, in continued proportion, and let the extremes of them A, D be prime to one another;

I say that A, B, C, D are the least of those which have the same ratio with them.

A B C D E F G H

For, if not, let E, F, G, H be less than A, B, C, D, and in the same ratio with them.

Now, since A, B, C, D are in the same ratio with E, F, G, H, and the multitude of the numbers A, B, C, D is equal to the multitude of the numbers E, F, G, H,
therefore, ex aequali, as A is to D, so is E to H. [VII. 14]
But A, D are prime, primes are also least, [VII. 21] and the least numbers measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent. [VII. 20]
Therefore A measures E, the greater the less : which is impossible.
Therefore E, F, G, H which are less than A, B, C, D are not in the same ratio with them.
Therefore A, B, C, D are the least of those which have the same ratio with them.
Q. E. D.

Dependency Graph

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"VII.13" [label="VII.13", fillcolor="#222244", tooltip="VII.13: If four numbers be proportional, they will also be proportional alternately.", URL="../../../book_vii/propositions/13/index.html", target="_top"]; "VII.14" [label="VII.14", fillcolor="#222244", tooltip="VII.14: If there be as many numbers as we please, and others equal to them in multitude, which taken two and two are in the same ratio, they will also be in the same ratio ex aequali.", URL="../../../book_vii/propositions/14/index.html", target="_top"]; "VII.15" [label="VII.15", fillcolor="#222244", tooltip="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…", URL="../../../book_vii/propositions/15/index.html", target="_top"]; "VII.16" [label="VII.16", fillcolor="#222244", tooltip="VII.16: If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.", URL="../../../book_vii/propositions/16/index.html", target="_top"]; 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"VII.2" -> "V.def.1"; "VII.15" -> "V.def.1"; "VII.20" -> "V.def.1"; "V.def.5" -> "V.def.2"; "VII.5" -> "V.def.2"; "V.def.17" -> "V.def.3"; "V.def.6" -> "V.def.5"; "VII.14" -> "V.def.5"; "VII.20" -> "V.def.5"; "VII.21" -> "V.def.5"; "VIII.1" -> "V.def.5"; "VII.13" -> "V.def.6"; "VIII.1" -> "V.def.6"; "V.def.12" -> "V.def.11"; "VII.12" -> "V.def.11"; "VII.9" -> "V.def.12"; "VII.10" -> "V.def.12"; "VII.13" -> "V.def.12"; "VII.15" -> "V.def.12"; "VII.14" -> "V.def.17"; "VII.def.2" -> "VII.def.1"; "VII.def.11" -> "VII.def.1"; "VII.def.12" -> "VII.def.1"; "VII.def.15" -> "VII.def.1"; "VII.1" -> "VII.def.1"; "VII.15" -> "VII.def.1"; "VII.def.3" -> "VII.def.2"; "VII.def.5" -> "VII.def.2"; "VII.def.15" -> "VII.def.2"; "VII.1" -> "VII.def.2"; "VII.2" -> "VII.def.2"; "VII.4" -> "VII.def.2"; "VII.5" -> "VII.def.2"; "VII.6" -> "VII.def.2"; "VII.9" -> "VII.def.2"; "VII.10" -> "VII.def.2"; "VII.12" -> "VII.def.2"; "VII.13" -> "VII.def.2"; "VII.14" -> "VII.def.2"; "VII.15" -> "VII.def.2"; "VII.16" -> "VII.def.2"; "VII.20" -> "VII.def.2"; "VII.21" -> "VII.def.2"; "VIII.1" -> "VII.def.2"; "VII.def.20" -> "VII.def.3"; "VII.4" -> "VII.def.3"; "VII.5" -> "VII.def.3"; "VII.9" -> "VII.def.3"; "VII.10" -> "VII.def.3"; "VII.def.20" -> "VII.def.4"; "VII.4" -> "VII.def.4"; "VII.6" -> "VII.def.4"; "VII.9" -> "VII.def.4"; "VII.10" -> "VII.def.4"; "VII.def.20" -> "VII.def.5"; "VII.5" -> "VII.def.5"; "VIII.1" -> "VII.def.11"; "VII.1" -> "VII.def.12"; "VII.2" -> "VII.def.12"; "VII.21" -> "VII.def.12"; "VIII.1" -> "VII.def.12"; "VII.16" -> "VII.def.15"; "VII.12" -> "VII.def.20"; "VII.13" -> "VII.def.20"; "VII.20" -> "VII.def.20"; "VII.2" -> "VII.1"; "VII.4" -> "VII.2"; "VII.20" -> "VII.4"; "VII.6" -> "VII.5"; "VII.9" -> "VII.5"; "VII.10" -> "VII.5"; "VII.12" -> "VII.5"; "VII.9" -> "VII.6"; "VII.10" -> "VII.6"; "VII.12" -> "VII.6"; "VII.10" -> "VII.9"; "VII.13" -> "VII.10"; "VII.15" -> "VII.12"; "VII.20" -> "VII.12"; "VII.14" -> "VII.13"; "VII.20" -> "VII.13"; "VIII.1" -> "VII.14"; "VII.16" -> "VII.15"; "VII.21" -> "VII.16"; "VII.21" -> "VII.20"; "VIII.1" -> "VII.20"; "VIII.1" -> "VII.21"; }

Required for

VIII.2, VIII.3, VIII.6, VIII.7, VIII.8, VIII.9, VIII.14, VIII.15, VIII.16, VIII.17, VIII.21, 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.12, IX.13, IX.15, IX.32, IX.36, X.9, X.10, X.29, X.30, X.31, X.32, X.33, X.34, X.35, X.39, X.40, X.41, X.45, X.46, X.48, X.49, X.50, X.51, X.52, X.53, X.57, X.58, X.59, X.60, X.63, X.64, X.65, X.68, X.69, X.70, X.71, X.72, X.76, X.77, X.78, X.82, X.83, X.84, X.85, X.86, X.87, X.88, X.89, X.90, X.91, X.94, X.95, X.96, X.100, X.101, X.102, X.105, X.106, X.107, X.108, X.109, X.110, X.111, XIII.6, XIII.11, XIII.16, XIII.17, XIII.18

See Commentary on Proposition VIII.1.