PROPOSITION 11.

Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate of that which the side has to the side.

Let A, B be square numbers, and let C be the side of A, and D of B;

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

For let C by multiplying D make E.

A B C D E
Now, since A is a square and C is its side,
therefore C by multiplying itself has made A.
For the same reason also
D by multiplying itself has made B.
Since then C by multiplying the numbers C, D has made A, E respectively,
[VII. 17]therefore, as C is to D, so is A to E.
For the same reason also,
[VII. 18]as C is to D, so is E to B.
Therefore also, as A is to E, so is E to B.
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 C has to D.

For, since A, E, B are three numbers in proportion,
therefore A has to B the ratio duplicate of that which A has to E. [V. Def. 9]
But, as A is to E, so is C to D.
Therefore A has to B the ratio duplicate of that which the side C has to D.
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"]; "I.def.8" [label="I.def.8", fillcolor="#224422", tooltip="I.def.8: A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-8", target="_top"]; "I.def.19" [label="I.def.19", fillcolor="#224422", tooltip="I.def.19: Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-19", target="_top"]; "I.def.20" [label="I.def.20", fillcolor="#224422", tooltip="I.def.20: Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-20", target="_top"]; "I.def.22" [label="I.def.22", fillcolor="#224422", tooltip="I.def.22: Of quadrilateral figures, a square is that which is both equilateral and right-angled ; an oblong that which is right-angled but not equilateral ; a rhombus that which is…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-22", 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.18" [label="VII.def.18", fillcolor="#224422", tooltip="VII.def.18: A square number is equal multiplied by equal, or a number which is contained by two equal numbers.", URL="../../../book_vii/definitions/index.html#vii-def-18", 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.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.12" [label="VII.12", fillcolor="#222244", tooltip="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.", URL="../../../book_vii/propositions/12/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.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"]; "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"]; "VII.18" [label="VII.18", fillcolor="#222244", tooltip="VII.18: If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.", URL="../../../book_vii/propositions/18/index.html", target="_top"]; "VIII.11" [label="VIII.11", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="VIII.11: Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate of that which the side has to the side.", 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.8" -> "I.def.2"; "I.def.4" -> "I.def.3"; "I.def.7" -> "I.def.4"; "I.def.8" -> "I.def.4"; "I.def.19" -> "I.def.4"; "I.def.7" -> "I.def.5"; "I.def.8" -> "I.def.7"; "I.def.22" -> "I.def.8"; "I.def.20" -> "I.def.19"; "I.def.22" -> "I.def.19"; "I.def.22" -> "I.def.20"; "VIII.11" -> "I.def.22"; "V.def.2" -> "V.def.1"; "VII.def.3" -> "V.def.1"; "VII.def.4" -> "V.def.1"; "VII.def.5" -> "V.def.1"; "VII.15" -> "V.def.1"; "V.def.5" -> "V.def.2"; "VII.5" -> "V.def.2"; "VIII.11" -> "V.def.3"; "V.def.6" -> "V.def.5"; "VII.17" -> "V.def.5"; "VII.18" -> "V.def.5"; "V.def.9" -> "V.def.6"; "VII.13" -> "V.def.6"; "VIII.11" -> "V.def.6"; "VIII.11" -> "V.def.9"; "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.def.2" -> "VII.def.1"; "VII.def.15" -> "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.def.18" -> "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.15" -> "VII.def.2"; "VII.16" -> "VII.def.2"; "VII.17" -> "VII.def.2"; "VII.18" -> "VII.def.2"; "VIII.11" -> "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.18" -> "VII.def.15"; "VII.16" -> "VII.def.15"; "VII.17" -> "VII.def.15"; "VII.18" -> "VII.def.15"; "VIII.11" -> "VII.def.18"; "VII.12" -> "VII.def.20"; "VII.13" -> "VII.def.20"; "VII.17" -> "VII.def.20"; "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.17" -> "VII.13"; "VII.16" -> "VII.15"; "VII.18" -> "VII.16"; "VII.18" -> "VII.17"; "VIII.11" -> "VII.17"; "VIII.11" -> "VII.18"; }

Required for

VIII.14, VIII.15, VIII.16, VIII.17, 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.11.