PROPOSITION 12.

Between two cube numbers there are two mean proportional numbers, and the cube has to the cube the ratio triplicate of that which the side has to the side.

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

I say that between A, B there are two mean proportional numbers, and A has to B the ratio triplicate of that which C has to D.

A B C D H K E F G

For let C by multiplying itself make E, and by multiplying D let it make F; let D by multiplying itself make G, and let the numbers C, D by multiplying F make H, K respectively.

Now, since A is a cube, and C its side, and C by multiplying itself has made E,
therefore C by multiplying itself has made E and by multiplying E has made A.
For the same reason also
D by multiplying itself has made G and by multiplying G has made B.
And, since C by multiplying the numbers C, D has made E, F respectively,
[VII. 17]therefore, as C is to D, so is E to F.
For the same reason also,
[VII. 18]as C is to D, so is F to G.
Again, since C by multiplying the numbers E, F has made A, H respectively,
[VII. 17]therefore, as E is to F, so is A to H.
But, as E is to F, so is C to D.
Therefore also, as C is to D, so is A to H.
Again, since the numbers C, D by multiplying F have made H, K respectively,
[VII. 18]therefore, as C is to D, so is H to K.
Again, since D by multiplying each of the numbers F, G has made K, B respectively,
[VII. 17]therefore, as F is to G, so is K to B.
But, as F is to G, so is C to D;
therefore also, as C is to D, so is A to H, H to K, and K to B.
Therefore H, K are two mean proportionals between A, B.

I say next that A also has to B the ratio triplicate of that which C has to D.

For, since A, H, K, B are four numbers in proportion,
therefore A has to B the ratio triplicate of that which A has to H. [V. Def. 10]
But, as A is to H, so is C to D;
therefore A also has to B the ratio triplicate of that which C has to D.
Q. E. D.

Dependency Graph

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"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.10" [label="V.def.10", fillcolor="#224422", tooltip="V.def.10: When four magnitudes are < continuously > proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually…", URL="../../../book_v/definitions/index.html#v-def-10", 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.19" [label="VII.def.19", fillcolor="#224422", tooltip="VII.def.19: And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.", URL="../../../book_vii/definitions/index.html#vii-def-19", 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.12" [label="VIII.12", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="VIII.12: Between two cube numbers there are two mean proportional numbers, and the cube has to the cube the ratio triplicate of that which the side has to the side.", URL="index.html", target="_top"]; "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.12" -> "V.def.3"; "V.def.6" -> "V.def.5"; "VII.17" -> "V.def.5"; "VII.18" -> "V.def.5"; "V.def.10" -> "V.def.6"; "VII.13" -> "V.def.6"; "VIII.12" -> "V.def.6"; "VIII.12" -> "V.def.10"; "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.19" -> "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.12" -> "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.19" -> "VII.def.15"; "VII.16" -> "VII.def.15"; "VII.17" -> "VII.def.15"; "VII.18" -> "VII.def.15"; "VIII.12" -> "VII.def.19"; "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.12" -> "VII.17"; "VIII.12" -> "VII.18"; }

Required for

VIII.15, VIII.17

See Commentary on Proposition VIII.12.