PROPOSITION 9.

If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be square, all the rest will also be square. And, if the number after the unit be cube, all the rest will also be cube.

Let there be as many numbers as we please, A, B, C, D, E, F, beginning from an unit and in continued proportion, and let A, the number after the unit, be square ;

I say that all the rest will also be square.

A B C D E F
Now it has been proved that B, the third from the unit, is square, as are also all those which leave out one ; [ix. 8]

I say that all the rest are also square.

For, since A, B, C are in continued proportion, and A is square,
[VIII. 22]therefore C is also square.
Again, since B, C, D are in continued proportion, and B is square,
[VIII. 22]D is also square.

Similarly we can prove that all the rest are also square.

Next, let A be cube ;

I say that all the rest are also cube.

Now it has been proved that C, the fourth from the unit, is cube, as also are all those which leave out two ; [ix. 8]

I say that all the rest are also cube.

For, since, as the unit is to A, so is A to B,
therefore the unit measures A the same number of times as A measures B.
But the unit measures A according to the units in it ;
therefore A also measures B according to the units in itself ;
therefore A by multiplying itself has made B.
And A is cube.
But, if a cube number by multiplying itself make some number, the product is cube. [IX. 3]
Therefore B is also cube.
And, since the four numbers A, B, C, D are in continued proportion, and A is cube,
[VIII. 23]D also is cube.
For the same reason
E is also cube, and similarly all the rest are cube.
Q. E. D.

Dependency Graph

digraph { bgcolor="transparent"; node [shape=box, style="rounded,filled", fontname="Helvetica", fontsize=11, color="#d6d3cd", fontcolor="#ffffff"]; edge [color="#8c857b", fontcolor="#8c857b", arrowsize=0.75]; rankdir="TB"; "I.def.1" [label="I.def.1", fillcolor="#224422", tooltip="I.def.1: A point is that which has no part.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-1", target="_top"]; "I.def.2" [label="I.def.2", fillcolor="#224422", tooltip="I.def.2: A line is breadthless length.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-2", target="_top"]; "I.def.3" [label="I.def.3", fillcolor="#224422", tooltip="I.def.3: The extremities of a line are points.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-3", target="_top"]; "I.def.4" [label="I.def.4", fillcolor="#224422", tooltip="I.def.4: A straight line is a line which lies evenly with the points on itself.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-4", target="_top"]; "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.13" [label="I.def.13", fillcolor="#224422", tooltip="I.def.13: A boundary is that which is an extremity of anything.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-13", target="_top"]; "I.def.14" [label="I.def.14", fillcolor="#224422", tooltip="I.def.14: A figure is that which is contained by any boundary or boundaries.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-14", 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.4" [label="V.def.4", fillcolor="#224422", tooltip="V.def.4: Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.", URL="../../../book_v/definitions/index.html#v-def-4", 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.7" [label="V.def.7", fillcolor="#224422", tooltip="V.def.7: When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then…", URL="../../../book_v/definitions/index.html#v-def-7", 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"]; "V.def.13" [label="V.def.13", fillcolor="#224422", tooltip="V.def.13: Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.", URL="../../../book_v/definitions/index.html#v-def-13", target="_top"]; "V.def.17" [label="V.def.17", fillcolor="#224422", tooltip="V.def.17: A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, as the first is to…", URL="../../../book_v/definitions/index.html#v-def-17", target="_top"]; "V.1" [label="V.1", fillcolor="#222244", tooltip="V.1: If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple one of the magnitudes is of one…", URL="../../../book_v/propositions/01/index.html", target="_top"]; "V.7" [label="V.7", fillcolor="#222244", tooltip="V.7: Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.", URL="../../../book_v/propositions/07/index.html", target="_top"]; "V.8" [label="V.8", fillcolor="#222244", tooltip="V.8: Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.", URL="../../../book_v/propositions/08/index.html", target="_top"]; "V.9" [label="V.9", fillcolor="#222244", tooltip="V.9: Magnitudes which have the same ratio to the same are equal to one another ; and magnitudes to which the same has the same ratio are equal.", URL="../../../book_v/propositions/09/index.html", 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.11" [label="VII.def.11", fillcolor="#224422", tooltip="VII.def.11: A prime number is that which is measured by an unit alone.", URL="../../../book_vii/definitions/index.html#vii-def-11", target="_top"]; "VII.def.12" [label="VII.def.12", fillcolor="#224422", tooltip="VII.def.12: Numbers prime to one another are those which are measured by an unit alone as a common measure.", URL="../../../book_vii/definitions/index.html#vii-def-12", 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.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.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.1" [label="VII.1", fillcolor="#222244", tooltip="VII.1: Two unequal numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit…", URL="../../../book_vii/propositions/01/index.html", target="_top"]; "VII.2" [label="VII.2", fillcolor="#222244", tooltip="VII.2: Given two numbers not prime to one another, to find their greatest common measure.", URL="../../../book_vii/propositions/02/index.html", target="_top"]; "VII.3" [label="VII.3", fillcolor="#222244", tooltip="VII.3: Given three numbers not prime to one another, to find their greatest common measure.", URL="../../../book_vii/propositions/03/index.html", target="_top"]; "VII.4" [label="VII.4", fillcolor="#222244", tooltip="VII.4: Any number is either a part or parts of any number, the less of the greater.", URL="../../../book_vii/propositions/04/index.html", 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.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"]; "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"]; "VII.19" [label="VII.19", fillcolor="#222244", tooltip="VII.19: If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the…", URL="../../../book_vii/propositions/19/index.html", target="_top"]; "VII.20" [label="VII.20", fillcolor="#222244", tooltip="VII.20: The least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less.", URL="../../../book_vii/propositions/20/index.html", target="_top"]; "VII.21" [label="VII.21", fillcolor="#222244", tooltip="VII.21: Numbers prime to one another are the least of those which have the same ratio with them.", URL="../../../book_vii/propositions/21/index.html", target="_top"]; "VII.22" [label="VII.22", fillcolor="#222244", tooltip="VII.22: The least numbers of those which have the same ratio with them are prime to one another.", URL="../../../book_vii/propositions/22/index.html", target="_top"]; "VII.23" [label="VII.23", fillcolor="#222244", tooltip="VII.23: If two numbers be prime to one another, the number which measures the one of them will be prime to the remaining number.", URL="../../../book_vii/propositions/23/index.html", target="_top"]; "VII.24" [label="VII.24", fillcolor="#222244", tooltip="VII.24: If two numbers be prime to any number, their product also will be prime to the same.", URL="../../../book_vii/propositions/24/index.html", target="_top"]; "VII.25" [label="VII.25", fillcolor="#222244", tooltip="VII.25: If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.", URL="../../../book_vii/propositions/25/index.html", target="_top"]; "VII.26" [label="VII.26", fillcolor="#222244", tooltip="VII.26: If two numbers be prime to two numbers, both to each, their products also will be prime to one another.", URL="../../../book_vii/propositions/26/index.html", target="_top"]; "VII.27" [label="VII.27", fillcolor="#222244", tooltip="VII.27: If two numbers be prime to one another, and each by multiplying itself make a certain number, the products will be prime to one another ; and, if the original numbers by…", URL="../../../book_vii/propositions/27/index.html", target="_top"]; "VII.33" [label="VII.33", fillcolor="#222244", tooltip="VII.33: Given as many numbers as we please, to find the least of those which have the same ratio with them.", URL="../../../book_vii/propositions/33/index.html", target="_top"]; "VIII.1" [label="VIII.1", fillcolor="#222244", tooltip="VIII.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…", URL="../../../book_viii/propositions/01/index.html", target="_top"]; "VIII.2" [label="VIII.2", fillcolor="#222244", tooltip="VIII.2: To find numbers in continued proportion, as many as may be prescribed, and the least that are in a given ratio.", URL="../../../book_viii/propositions/02/index.html", target="_top"]; "VIII.3" [label="VIII.3", fillcolor="#222244", tooltip="VIII.3: If as many numbers as we please in continued proportion be the least of those which have the same ratio with them, the extremes of them are prime to one another.", URL="../../../book_viii/propositions/03/index.html", target="_top"]; "VIII.8" [label="VIII.8", fillcolor="#222244", tooltip="VIII.8: If between two numbers there fall numbers in continued proportion with them, then, however many numbers fall between them in continued proportion, so many will also fall in…", URL="../../../book_viii/propositions/08/index.html", target="_top"]; "VIII.20" [label="VIII.20", fillcolor="#222244", tooltip="VIII.20: If one mean proportional number fall between two numbers, the numbers will be similar plane numbers.", URL="../../../book_viii/propositions/20/index.html", target="_top"]; "VIII.21" [label="VIII.21", fillcolor="#222244", tooltip="VIII.21: If two mean proportional numbers fall between two numbers, the numbers are similar solid numbers.", URL="../../../book_viii/propositions/21/index.html", target="_top"]; "VIII.22" [label="VIII.22", fillcolor="#222244", tooltip="VIII.22: If three numbers be in continued proportion, and the first be square, the third will also be square.", URL="../../../book_viii/propositions/22/index.html", target="_top"]; "VIII.23" [label="VIII.23", fillcolor="#222244", tooltip="VIII.23: If four numbers be in continued proportion, and the first be cube, the fourth will also be cube.", URL="../../../book_viii/propositions/23/index.html", target="_top"]; "IX.3" [label="IX.3", fillcolor="#222244", tooltip="IX.3: If a cube number by multiplying itself make some number, the product will be cube.", URL="../03/index.html", target="_top"]; "IX.8" [label="IX.8", fillcolor="#222244", tooltip="IX.8: If as many numbers as we please beginning from an unit be in continued proportion, the third from the unit will be square, as will also those which successively leave out one; the…", URL="../08/index.html", target="_top"]; "IX.9" [label="IX.9", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="IX.9: If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be square, all the rest will also be square. And, if the number…", 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.13" -> "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"; "VIII.20" -> "I.def.7"; "I.def.22" -> "I.def.8"; "I.def.14" -> "I.def.13"; "V.9" -> "I.def.14"; "I.def.20" -> "I.def.19"; "I.def.22" -> "I.def.19"; "I.def.22" -> "I.def.20"; "VIII.2" -> "I.def.22"; "VIII.22" -> "I.def.22"; "IX.8" -> "I.def.22"; "IX.9" -> "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.def.11" -> "V.def.1"; "VII.def.12" -> "V.def.1"; "VII.1" -> "V.def.1"; "VII.2" -> "V.def.1"; "VII.3" -> "V.def.1"; "VII.15" -> "V.def.1"; "VII.20" -> "V.def.1"; "VII.23" -> "V.def.1"; "V.def.5" -> "V.def.2"; "V.def.7" -> "V.def.2"; "V.1" -> "V.def.2"; "VII.5" -> "V.def.2"; "V.def.17" -> "V.def.3"; "VII.33" -> "V.def.3"; "VIII.2" -> "V.def.3"; "V.8" -> "V.def.4"; "V.def.6" -> "V.def.5"; "V.7" -> "V.def.5"; "V.9" -> "V.def.5"; "VII.14" -> "V.def.5"; "VII.17" -> "V.def.5"; "VII.18" -> "V.def.5"; "VII.20" -> "V.def.5"; "VII.21" -> "V.def.5"; "VII.22" -> "V.def.5"; "VII.33" -> "V.def.5"; "VIII.1" -> "V.def.5"; "VIII.3" -> "V.def.5"; "VIII.8" -> "V.def.5"; "V.7" -> "V.def.6"; "VII.def.21" -> "V.def.6"; "VII.13" -> "V.def.6"; "VII.19" -> "V.def.6"; "VIII.1" -> "V.def.6"; "VIII.2" -> "V.def.6"; "VIII.3" -> "V.def.6"; "VIII.8" -> "V.def.6"; "VIII.20" -> "V.def.6"; "VIII.21" -> "V.def.6"; "VIII.22" -> "V.def.6"; "VIII.23" -> "V.def.6"; "IX.8" -> "V.def.6"; "IX.9" -> "V.def.6"; "V.8" -> "V.def.7"; "V.def.12" -> "V.def.11"; "V.def.13" -> "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"; "V.7" -> "V.def.13"; "VII.14" -> "V.def.17"; "V.8" -> "V.1"; "VII.19" -> "V.7"; "V.9" -> "V.8"; "VII.19" -> "V.9"; "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"; "IX.8" -> "VII.def.1"; "IX.9" -> "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.def.19" -> "VII.def.2"; "VII.1" -> "VII.def.2"; "VII.2" -> "VII.def.2"; "VII.3" -> "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.17" -> "VII.def.2"; "VII.18" -> "VII.def.2"; "VII.19" -> "VII.def.2"; "VII.20" -> "VII.def.2"; "VII.21" -> "VII.def.2"; "VII.22" -> "VII.def.2"; "VII.23" -> "VII.def.2"; "VII.24" -> "VII.def.2"; "VII.25" -> "VII.def.2"; "VII.26" -> "VII.def.2"; "VII.27" -> "VII.def.2"; "VII.33" -> "VII.def.2"; "VIII.1" -> "VII.def.2"; "VIII.2" -> "VII.def.2"; "VIII.3" -> "VII.def.2"; "VIII.8" -> "VII.def.2"; "VIII.20" -> "VII.def.2"; "VIII.21" -> "VII.def.2"; "VIII.22" -> "VII.def.2"; "VIII.23" -> "VII.def.2"; "IX.3" -> "VII.def.2"; "IX.8" -> "VII.def.2"; "IX.9" -> "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.3" -> "VII.def.12"; "VII.21" -> "VII.def.12"; "VII.22" -> "VII.def.12"; "VII.23" -> "VII.def.12"; "VII.24" -> "VII.def.12"; "VII.25" -> "VII.def.12"; "VII.26" -> "VII.def.12"; "VII.27" -> "VII.def.12"; "VIII.1" -> "VII.def.12"; "VIII.3" -> "VII.def.12"; "VII.def.16" -> "VII.def.15"; "VII.def.19" -> "VII.def.15"; "VII.16" -> "VII.def.15"; "VII.17" -> "VII.def.15"; "VII.18" -> "VII.def.15"; "VII.22" -> "VII.def.15"; "VII.24" -> "VII.def.15"; "VII.27" -> "VII.def.15"; "VII.33" -> "VII.def.15"; "IX.3" -> "VII.def.15"; "VIII.20" -> "VII.def.16"; "IX.3" -> "VII.def.19"; "VII.12" -> "VII.def.20"; "VII.13" -> "VII.def.20"; "VII.17" -> "VII.def.20"; "VII.20" -> "VII.def.20"; "VII.33" -> "VII.def.20"; "VIII.8" -> "VII.def.20"; "IX.3" -> "VII.def.20"; "IX.8" -> "VII.def.20"; "VIII.20" -> "VII.def.21"; "VIII.21" -> "VII.def.21"; "VII.2" -> "VII.1"; "VII.3" -> "VII.2"; "VII.4" -> "VII.2"; "VII.33" -> "VII.3"; "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.17" -> "VII.13"; "VII.20" -> "VII.13"; "VIII.20" -> "VII.13"; "VIII.1" -> "VII.14"; "VIII.8" -> "VII.14"; "VIII.21" -> "VII.14"; "VII.16" -> "VII.15"; "VII.18" -> "VII.16"; "VII.21" -> "VII.16"; "VII.24" -> "VII.16"; "VII.33" -> "VII.16"; "VII.18" -> "VII.17"; "VII.19" -> "VII.17"; "VII.22" -> "VII.17"; "VIII.2" -> "VII.17"; "VIII.20" -> "VII.17"; "VII.19" -> "VII.18"; "VIII.2" -> "VII.18"; "VIII.21" -> "VII.18"; "VII.24" -> "VII.19"; "VII.33" -> "VII.19"; "VII.21" -> "VII.20"; "VII.24" -> "VII.20"; "VIII.1" -> "VII.20"; "VIII.8" -> "VII.20"; "VIII.20" -> "VII.20"; "VIII.21" -> "VII.20"; "VII.24" -> "VII.21"; "VII.33" -> "VII.21"; "VIII.1" -> "VII.21"; "VIII.8" -> "VII.21"; "VIII.21" -> "VII.21"; "VIII.2" -> "VII.22"; "VIII.3" -> "VII.22"; "VII.24" -> "VII.23"; "VII.25" -> "VII.24"; "VII.26" -> "VII.24"; "VII.27" -> "VII.25"; "VII.27" -> "VII.26"; "VIII.2" -> "VII.27"; "VIII.3" -> "VII.27"; "VIII.3" -> "VII.33"; "VIII.8" -> "VII.33"; "VIII.20" -> "VII.33"; "VIII.21" -> "VII.33"; "VIII.2" -> "VIII.1"; "VIII.3" -> "VIII.2"; "VIII.21" -> "VIII.2"; "VIII.8" -> "VIII.3"; "VIII.21" -> "VIII.3"; "IX.3" -> "VIII.8"; "VIII.21" -> "VIII.20"; "VIII.22" -> "VIII.20"; "VIII.23" -> "VIII.21"; "IX.8" -> "VIII.22"; "IX.9" -> "VIII.22"; "IX.3" -> "VIII.23"; "IX.8" -> "VIII.23"; "IX.9" -> "VIII.23"; "IX.9" -> "IX.3"; "IX.9" -> "IX.8"; }

See Commentary on Proposition IX.9.