PROPOSITION 18.

Between two similar plane numbers there is one mean proportional number; and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side.

Let A, B be two similar plane numbers, and let the numbers C, D be the sides of A, and E, F of B.

A B G C D E F
Now, since similar plane numbers are those which have their sides proportional, [VII. Def. 21]
therefore, as C is to D, so is E to F.

I say then that between A, B there is one mean proportional number, and A has to B the ratio duplicate of that which C has to E, or D to F, that is, of that which the corresponding side has to the corresponding side.

Now since, as C is to D, so is E to F,
[VII. 13]therefore, alternately, as C is to E, so is D to F.
And, since A is plane, and C, D are its sides,
therefore D by multiplying C has made A.
For the same reason also
E by multiplying F has made B.

Now let D by multiplying E make G.

Then, since D by multiplying C has made A, and by multiplying E has made G,
[VII. 17]therefore, as C is to E, so is A to G.
But, as C is to E, so is D to F;
therefore also, as D is to F, so is A to G.
Again, since E by multiplying D has made G, and by multiplying F has made B,
[VII. 17]therefore, as D is to F, so is G to B.
But it was also proved that, as D is to F, so is A to G;
therefore also, as A is to G, so is G to B.
Therefore A, G, B are in continued proportion.
Therefore between A, B there is one mean proportional number.

I say next that A also has to B the ratio duplicate of that which the corresponding side has to the corresponding side, that is, of that which C has to E or D to F.

For, since A, G, B are in continued proportion,
[V. Def. 9]A has to B the ratio duplicate of that which it has to G.
And, as A is to G, so is C to E, and so is D to F.
Therefore A also has to B the ratio duplicate of that which C has to E or D to F.
Q. E. D.

Dependency Graph

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"I.def.5" [label="I.def.5", fillcolor="#224422", tooltip="I.def.5: A surface is that which has length and breadth only.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-5", target="_top"]; "I.def.7" [label="I.def.7", fillcolor="#224422", tooltip="I.def.7: A plane surface is a surface which lies evenly with the straight lines on itself.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-7", target="_top"]; "V.def.1" [label="V.def.1", fillcolor="#224422", tooltip="V.def.1: A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.", URL="../../../book_v/definitions/index.html#v-def-1", target="_top"]; "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.3" [label="V.def.3", fillcolor="#224422", tooltip="V.def.3: A ratio is a sort of relation in respect of size between two magnitudes of the same kind.", URL="../../../book_v/definitions/index.html#v-def-3", target="_top"]; "V.def.5" [label="V.def.5", fillcolor="#224422", tooltip="V.def.5: Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any…", URL="../../../book_v/definitions/index.html#v-def-5", target="_top"]; "V.def.6" [label="V.def.6", fillcolor="#224422", tooltip="V.def.6: Let magnitudes which have the same ratio be called proportional.", URL="../../../book_v/definitions/index.html#v-def-6", target="_top"]; "V.def.9" [label="V.def.9", fillcolor="#224422", tooltip="V.def.9: When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second.", URL="../../../book_v/definitions/index.html#v-def-9", 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="../../../book_vii/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="../../../book_vii/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="../../../book_vii/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="../../../book_vii/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="../../../book_vii/definitions/index.html#vii-def-5", target="_top"]; "VII.def.15" [label="VII.def.15", fillcolor="#224422", tooltip="VII.def.15: A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.", URL="../../../book_vii/definitions/index.html#vii-def-15", target="_top"]; "VII.def.16" [label="VII.def.16", fillcolor="#224422", tooltip="VII.def.16: And, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.", URL="../../../book_vii/definitions/index.html#vii-def-16", 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="../../../book_vii/definitions/index.html#vii-def-20", target="_top"]; "VII.def.21" [label="VII.def.21", fillcolor="#224422", tooltip="VII.def.21: Similar plane and solid numbers are those which have their sides proportional.", URL="../../../book_vii/definitions/index.html#vii-def-21", 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="../../../book_vii/propositions/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="../../../book_vii/propositions/06/index.html", target="_top"]; "VII.9" [label="VII.9", fillcolor="#222244", tooltip="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…", URL="../../../book_vii/propositions/09/index.html", target="_top"]; "VII.10" [label="VII.10", fillcolor="#222244", tooltip="VII.10: If a number be parts of a number, and another be the same parts of another, alternately also, whatever parts or part the first is of the third, the same parts or the same part…", URL="../../../book_vii/propositions/10/index.html", target="_top"]; "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.17" [label="VII.17", fillcolor="#222244", tooltip="VII.17: If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.", URL="../../../book_vii/propositions/17/index.html", target="_top"]; "VIII.18" [label="VIII.18", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="VIII.18: Between two similar plane numbers there is one mean proportional number; and the plane number has to the plane number the ratio duplicate of that which the corresponding side has…", URL="index.html", target="_top"]; "I.def.3" -> "I.def.1"; "I.def.4" -> "I.def.1"; "I.def.3" -> "I.def.2"; "I.def.4" -> "I.def.2"; "I.def.4" -> "I.def.3"; "I.def.7" -> "I.def.4"; "I.def.7" -> "I.def.5"; "VIII.18" -> "I.def.7"; "V.def.2" -> "V.def.1"; "VII.def.3" -> "V.def.1"; "VII.def.4" -> "V.def.1"; "VII.def.5" -> "V.def.1"; "V.def.5" -> "V.def.2"; "VII.5" -> "V.def.2"; "VIII.18" -> "V.def.3"; "V.def.6" -> "V.def.5"; "VII.17" -> "V.def.5"; "V.def.9" -> "V.def.6"; "VII.def.21" -> "V.def.6"; "VII.13" -> "V.def.6"; "VIII.18" -> "V.def.6"; "VIII.18" -> "V.def.9"; "V.def.12" -> "V.def.11"; "VII.9" -> "V.def.12"; "VII.10" -> "V.def.12"; "VII.13" -> "V.def.12"; "VII.def.2" -> "VII.def.1"; "VII.def.15" -> "VII.def.1"; "VII.def.3" -> "VII.def.2"; "VII.def.5" -> "VII.def.2"; "VII.def.15" -> "VII.def.2"; "VII.def.16" -> "VII.def.2"; "VII.5" -> "VII.def.2"; "VII.6" -> "VII.def.2"; "VII.9" -> "VII.def.2"; "VII.10" -> "VII.def.2"; "VII.13" -> "VII.def.2"; "VII.17" -> "VII.def.2"; "VIII.18" -> "VII.def.2"; "VII.def.20" -> "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.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"; "VII.def.16" -> "VII.def.15"; "VII.17" -> "VII.def.15"; "VIII.18" -> "VII.def.16"; "VII.13" -> "VII.def.20"; "VII.17" -> "VII.def.20"; "VIII.18" -> "VII.def.21"; "VII.6" -> "VII.5"; "VII.9" -> "VII.5"; "VII.10" -> "VII.5"; "VII.9" -> "VII.6"; "VII.10" -> "VII.6"; "VII.10" -> "VII.9"; "VII.13" -> "VII.10"; "VII.17" -> "VII.13"; "VIII.18" -> "VII.13"; "VIII.18" -> "VII.17"; }

Required for

VIII.19, VIII.24, VIII.25, VIII.26, VIII.27, IX.1, IX.2, IX.4, IX.5, IX.6, IX.10, 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.18.