PROPOSITION 65.

The square on the side of the sum of two medial areas applied to a rational straight line produces as breadth the sixth binomial.

Let AB be the side of the sum of two medial areas, divided at C, let DE be a rational straight line, and let there be applied to DE the parallelogram DF equal to the square on AB, producing DG as its breadth ;

I say that DG is a sixth binomial straight line.

For let the same construction be made as before.

Then, since AB is the side of the sum of two medial areas, divided at C,
D K M N G E H L O F A C B
therefore AC, CB are straight lines incommensurable in square which make the sum of the squares on them medial, the rectangle contained by them medial, and moreover the sum of the squares on them incommensurable with the rectangle contained by them, [X. 41] so that, in accordance with what was before proved, each of the rectangles DL, MF is medial.
And they are applied to the rational straight line DE;
therefore each of the straight lines DM, MG is rational and incommensurable in length with DE. [X. 22]
And, since the sum of the squares on AC, CB is incommensurable with twice the rectangle AC, CB,
therefore DL is incommensurable with MF.
Therefore DM is also incommensurable with MG; [VI. 1, X. 11] therefore DM, MG are rational straight lines commensurable in square only; therefore DG is binomial. [X. 36]

I say next that it is also a sixth binomial straight line.

Similarly again we can prove that the rectangle DK, KM is equal to the square on MN, and that DK is incommensurable in length with KM; and, for the same reason, the square on DM is greater than the square on MG by the square on a straight line incommensurable in length with DM.
And neither of the straight lines DM, MG is commensurable in length with the rational straight line DE set out.
Therefore DG is a sixth binomial straight line. Q. E. D.

Dependency Graph

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And the centre of the semicircle is the same as that of the circle.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-18", 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.21" [label="I.def.21", fillcolor="#224422", tooltip="I.def.21: Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-21", 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"]; "I.def.23" [label="I.def.23", fillcolor="#224422", tooltip="I.def.23: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-23", target="_top"]; "I.post.1" [label="I.post.1", fillcolor="#444422", tooltip="I.post.1: To draw a straight line from any point to any point.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-1", target="_top"]; "I.post.2" [label="I.post.2", fillcolor="#444422", tooltip="I.post.2: To produce a finite straight line continuously in a straight line.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-2", target="_top"]; "I.post.3" [label="I.post.3", fillcolor="#444422", tooltip="I.post.3: To describe a circle with any centre and distance.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-3", target="_top"]; "I.post.4" [label="I.post.4", fillcolor="#444422", tooltip="I.post.4: That all right angles are equal to one another.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-4", target="_top"]; "I.post.5" [label="I.post.5", fillcolor="#444422", tooltip="I.post.5: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet…", URL="../../../../volume_I/book_i/postulates/index.html#i-post-5", target="_top"]; "I.cn.1" [label="I.cn.1", fillcolor="#442222", tooltip="I.cn.1: Things which are equal to the same thing are also equal to one another.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-1", target="_top"]; "I.cn.2" [label="I.cn.2", fillcolor="#442222", tooltip="I.cn.2: If equals be added to equals, the wholes are equal.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-2", target="_top"]; "I.cn.3" [label="I.cn.3", fillcolor="#442222", tooltip="I.cn.3: If equals be subtracted from equals, the remainders are equal.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-3", target="_top"]; "I.cn.4" [label="I.cn.4", fillcolor="#442222", tooltip="I.cn.4: Things which coincide with one another are equal to one another.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-4", target="_top"]; "I.cn.5" [label="I.cn.5", fillcolor="#442222", tooltip="I.cn.5: The whole is greater than the part.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-5", target="_top"]; "I.1" [label="I.1", fillcolor="#222244", tooltip="I.1: On a given finite straight line to construct an equilateral triangle.", URL="../../../../volume_I/book_i/propositions/01/index.html", target="_top"]; "I.2" [label="I.2", fillcolor="#222244", tooltip="I.2: To place at a given point (as an extremity) a straight line equal to a given straight line.", URL="../../../../volume_I/book_i/propositions/02/index.html", target="_top"]; "I.3" [label="I.3", fillcolor="#222244", tooltip="I.3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less.", URL="../../../../volume_I/book_i/propositions/03/index.html", target="_top"]; "I.4" [label="I.4", fillcolor="#222244", tooltip="I.4: If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the…", URL="../../../../volume_I/book_i/propositions/04/index.html", target="_top"]; "I.5" [label="I.5", fillcolor="#222244", tooltip="I.5: In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one…", URL="../../../../volume_I/book_i/propositions/05/index.html", target="_top"]; "I.6" [label="I.6", fillcolor="#222244", tooltip="I.6: If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another.", URL="../../../../volume_I/book_i/propositions/06/index.html", target="_top"]; "I.7" [label="I.7", fillcolor="#222244", tooltip="I.7: Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be constructed on the same straight line (from its…", URL="../../../../volume_I/book_i/propositions/07/index.html", target="_top"]; "I.8" [label="I.8", fillcolor="#222244", tooltip="I.8: If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the…", URL="../../../../volume_I/book_i/propositions/08/index.html", target="_top"]; "I.9" [label="I.9", fillcolor="#222244", tooltip="I.9: To bisect a given rectilineal angle.", URL="../../../../volume_I/book_i/propositions/09/index.html", target="_top"]; "I.10" [label="I.10", fillcolor="#222244", tooltip="I.10: To bisect a given finite straight line.", URL="../../../../volume_I/book_i/propositions/10/index.html", target="_top"]; "I.11" [label="I.11", fillcolor="#222244", tooltip="I.11: To draw a straight line at right angles to a given straight line from a given point on it.", URL="../../../../volume_I/book_i/propositions/11/index.html", target="_top"]; "I.13" [label="I.13", fillcolor="#222244", tooltip="I.13: If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles.", URL="../../../../volume_I/book_i/propositions/13/index.html", target="_top"]; "I.14" [label="I.14", fillcolor="#222244", tooltip="I.14: If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be…", URL="../../../../volume_I/book_i/propositions/14/index.html", target="_top"]; "I.15" [label="I.15", fillcolor="#222244", tooltip="I.15: If two straight lines cut one another, they make the vertical angles equal to one another.", URL="../../../../volume_I/book_i/propositions/15/index.html", target="_top"]; "I.16" [label="I.16", fillcolor="#222244", tooltip="I.16: In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles.", URL="../../../../volume_I/book_i/propositions/16/index.html", target="_top"]; "I.17" [label="I.17", fillcolor="#222244", tooltip="I.17: In any triangle two angles taken together in any manner are less than two right angles.", URL="../../../../volume_I/book_i/propositions/17/index.html", target="_top"]; "I.18" [label="I.18", fillcolor="#222244", tooltip="I.18: In any triangle the greater side subtends the greater angle.", URL="../../../../volume_I/book_i/propositions/18/index.html", target="_top"]; "I.19" [label="I.19", fillcolor="#222244", tooltip="I.19: In any triangle the greater angle is subtended by the greater side.", URL="../../../../volume_I/book_i/propositions/19/index.html", target="_top"]; "I.20" [label="I.20", fillcolor="#222244", tooltip="I.20: In any triangle two sides taken together in any manner are greater than the remaining one.", URL="../../../../volume_I/book_i/propositions/20/index.html", target="_top"]; "I.22" [label="I.22", fillcolor="#222244", tooltip="I.22: Out of three straight lines, which are equal to three given straight lines, to construct a triangle: thus it is necessary that two of the straight lines taken together in any…", URL="../../../../volume_I/book_i/propositions/22/index.html", target="_top"]; "I.23" [label="I.23", fillcolor="#222244", tooltip="I.23: On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.", URL="../../../../volume_I/book_i/propositions/23/index.html", target="_top"]; "I.26" [label="I.26", fillcolor="#222244", tooltip="I.26: If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one…", URL="../../../../volume_I/book_i/propositions/26/index.html", target="_top"]; "I.27" [label="I.27", fillcolor="#222244", tooltip="I.27: If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.", URL="../../../../volume_I/book_i/propositions/27/index.html", target="_top"]; "I.28" [label="I.28", fillcolor="#222244", tooltip="I.28: If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal…", URL="../../../../volume_I/book_i/propositions/28/index.html", target="_top"]; "I.29" [label="I.29", fillcolor="#222244", tooltip="I.29: A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior…", URL="../../../../volume_I/book_i/propositions/29/index.html", target="_top"]; "I.31" [label="I.31", fillcolor="#222244", tooltip="I.31: Through a given point to draw a straight line parallel to a given straight line.", URL="../../../../volume_I/book_i/propositions/31/index.html", target="_top"]; "I.32" [label="I.32", fillcolor="#222244", tooltip="I.32: In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to…", URL="../../../../volume_I/book_i/propositions/32/index.html", target="_top"]; "I.33" [label="I.33", fillcolor="#222244", tooltip="I.33: The straight lines joining equal and parallel straight lines (at the extremities which are) in the same directions (respectively) are themselves also equal and parallel.", URL="../../../../volume_I/book_i/propositions/33/index.html", target="_top"]; "I.34" [label="I.34", fillcolor="#222244", tooltip="I.34: In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.", URL="../../../../volume_I/book_i/propositions/34/index.html", target="_top"]; "I.35" [label="I.35", fillcolor="#222244", tooltip="I.35: Parallelograms which are on the same base and in the same parallels are equal to one another.", URL="../../../../volume_I/book_i/propositions/35/index.html", target="_top"]; "I.36" [label="I.36", fillcolor="#222244", tooltip="I.36: Parallelograms which are on equal bases and in the same parallels are equal to one another.", URL="../../../../volume_I/book_i/propositions/36/index.html", target="_top"]; "I.37" [label="I.37", fillcolor="#222244", tooltip="I.37: Triangles which are on the same base and in the same parallels are equal to one another.", URL="../../../../volume_I/book_i/propositions/37/index.html", target="_top"]; "I.38" [label="I.38", fillcolor="#222244", tooltip="I.38: Triangles which are on equal bases and in the same parallels are equal to one another.", URL="../../../../volume_I/book_i/propositions/38/index.html", target="_top"]; "I.39" [label="I.39", fillcolor="#222244", tooltip="I.39: Equal triangles which are on the same base and on the same side are also in the same parallels.", URL="../../../../volume_I/book_i/propositions/39/index.html", target="_top"]; "I.41" [label="I.41", fillcolor="#222244", tooltip="I.41: If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle.", URL="../../../../volume_I/book_i/propositions/41/index.html", target="_top"]; "I.43" [label="I.43", fillcolor="#222244", tooltip="I.43: In any parallelogram the complements of the parallelograms about the diameter are equal to one another.", URL="../../../../volume_I/book_i/propositions/43/index.html", target="_top"]; "I.46" [label="I.46", fillcolor="#222244", tooltip="I.46: On a given straight line to describe a square.", URL="../../../../volume_I/book_i/propositions/46/index.html", target="_top"]; "I.47" [label="I.47", fillcolor="#222244", tooltip="I.47: In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.", URL="../../../../volume_I/book_i/propositions/47/index.html", target="_top"]; "II.def.1" [label="II.def.1", fillcolor="#224422", tooltip="II.def.1: Any rectangular parallelogram is said to be contained by the two straight lines containing the right angle.", URL="../../../../volume_I/book_ii/definitions/index.html#ii-def-1", target="_top"]; "II.def.2" [label="II.def.2", fillcolor="#224422", tooltip="II.def.2: And in any parallelogrammic area let any one whatever of the parallelograms about its diameter with the two complements be called a gnomon.", URL="../../../../volume_I/book_ii/definitions/index.html#ii-def-2", target="_top"]; "II.4" [label="II.4", fillcolor="#222244", tooltip="II.4: If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.", URL="../../../../volume_I/book_ii/propositions/04/index.html", target="_top"]; "II.5" [label="II.5", fillcolor="#222244", tooltip="II.5: If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the…", URL="../../../../volume_I/book_ii/propositions/05/index.html", target="_top"]; "III.def.8" [label="III.def.8", fillcolor="#224422", tooltip="III.def.8: An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line…", URL="../../../../volume_II/book_iii/definitions/index.html#iii-def-8", target="_top"]; "III.def.10" [label="III.def.10", fillcolor="#224422", tooltip="III.def.10: A sector of a circle is the figure which, when an angle is constructed at the centre of the circle, is contained by the straight lines containing the angle and the circumference…", URL="../../../../volume_II/book_iii/definitions/index.html#iii-def-10", target="_top"]; "III.20" [label="III.20", fillcolor="#222244", tooltip="III.20: In a circle the angle at the centre is double of the angle at the circumference, when the angles have the same circumference as base.", URL="../../../../volume_II/book_iii/propositions/20/index.html", target="_top"]; "III.21" [label="III.21", fillcolor="#222244", tooltip="III.21: In a circle the angles in the same segment are equal to one another.", URL="../../../../volume_II/book_iii/propositions/21/index.html", target="_top"]; "III.22" [label="III.22", fillcolor="#222244", tooltip="III.22: The opposite angles of quadrilaterals in circles are equal to two right angles.", URL="../../../../volume_II/book_iii/propositions/22/index.html", target="_top"]; "III.31" [label="III.31", fillcolor="#222244", tooltip="III.31: In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; and further the angle…", URL="../../../../volume_II/book_iii/propositions/31/index.html", target="_top"]; "IV.def.7" [label="IV.def.7", fillcolor="#224422", tooltip="IV.def.7: A straight line is said to be fitted into a circle when its extremities are on the circumference of the circle.", URL="../../../../volume_II/book_iv/definitions/index.html#iv-def-7", target="_top"]; "IV.1" [label="IV.1", fillcolor="#222244", tooltip="IV.1: Into a given circle to fit a straight line equal to a given straight line which is not greater than the diameter of the circle.", URL="../../../../volume_II/book_iv/propositions/01/index.html", 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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/book_v/definitions/index.html#v-def-7", 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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/book_v/definitions/index.html#v-def-13", target="_top"]; "V.def.14" [label="V.def.14", fillcolor="#224422", tooltip="V.def.14: Composition of a ratio means taking the antecedent together with the consequent as one in relation to the consequent by itself.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-14", target="_top"]; "V.def.15" [label="V.def.15", fillcolor="#224422", tooltip="V.def.15: Separation of a ratio means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-15", target="_top"]; "V.def.16" [label="V.def.16", fillcolor="#224422", tooltip="V.def.16: Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-16", 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="../../../../volume_II/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="../../../../volume_II/book_v/propositions/01/index.html", target="_top"]; "V.2" [label="V.2", fillcolor="#222244", tooltip="V.2: If a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same multiple of the second that a sixth is of the fourth, the sum of…", URL="../../../../volume_II/book_v/propositions/02/index.html", target="_top"]; "V.3" [label="V.3", fillcolor="#222244", tooltip="V.3: If a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the first and third, then also ex aequali the magnitudes taken…", URL="../../../../volume_II/book_v/propositions/03/index.html", target="_top"]; "V.4" [label="V.4", fillcolor="#222244", tooltip="V.4: If a first magnitude have to a second the same ratio as a third to a fourth, any equimultiples whatever of the first and third will also have the same ratio to any equimultiples…", URL="../../../../volume_II/book_v/propositions/04/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/book_v/propositions/09/index.html", target="_top"]; "V.10" [label="V.10", fillcolor="#222244", tooltip="V.10: Of magnitudes which have a ratio to the same, that which has a greater ratio is greater ; and that to which the same has a greater ratio is less.", URL="../../../../volume_II/book_v/propositions/10/index.html", target="_top"]; "V.11" [label="V.11", fillcolor="#222244", tooltip="V.11: Ratios which are the same with the same ratio are also the same with one another.", URL="../../../../volume_II/book_v/propositions/11/index.html", target="_top"]; "V.12" [label="V.12", fillcolor="#222244", tooltip="V.12: If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.", URL="../../../../volume_II/book_v/propositions/12/index.html", target="_top"]; "V.13" [label="V.13", fillcolor="#222244", tooltip="V.13: If a first magnitude have to a second the same ratio as a third to a fourth, and the third have to the fourth a greater ratio than a fifth has to a sixth, the first will also have…", URL="../../../../volume_II/book_v/propositions/13/index.html", target="_top"]; "V.14" [label="V.14", fillcolor="#222244", tooltip="V.14: If a first magnitude have to a second the same ratio as a third has to a fourth, and the first be greater than the third, the second will also be greater than the fourth ; if…", URL="../../../../volume_II/book_v/propositions/14/index.html", target="_top"]; "V.15" [label="V.15", fillcolor="#222244", tooltip="V.15: Parts have the same ratio as the same multiples of them taken in corresponding order.", URL="../../../../volume_II/book_v/propositions/15/index.html", target="_top"]; "V.16" [label="V.16", fillcolor="#222244", tooltip="V.16: If four magnitudes be proportional, they will also be proportional alternately.", URL="../../../../volume_II/book_v/propositions/16/index.html", target="_top"]; "V.17" [label="V.17", fillcolor="#222244", tooltip="V.17: If magnitudes be proportional componendo, they will also be proportional separando.", URL="../../../../volume_II/book_v/propositions/17/index.html", target="_top"]; "V.19" [label="V.19", fillcolor="#222244", tooltip="V.19: If, as a whole is to a whole, so is a part subtracted to a part subtracted, the remainder will also be to the remainder as whole to whole.", URL="../../../../volume_II/book_v/propositions/19/index.html", target="_top"]; "V.20" [label="V.20", fillcolor="#222244", tooltip="V.20: If there be three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first be greater than the third, the…", URL="../../../../volume_II/book_v/propositions/20/index.html", target="_top"]; "V.22" [label="V.22", fillcolor="#222244", tooltip="V.22: If there be any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, they will also be in the same ratio…", URL="../../../../volume_II/book_v/propositions/22/index.html", target="_top"]; "VI.def.1" [label="VI.def.1", fillcolor="#224422", tooltip="VI.def.1: Similar rectilineal figures are such as have their angles severally equal and the sides about the equal angles proportional.", URL="../../../../volume_II/book_vi/definitions/index.html#vi-def-1", target="_top"]; "VI.def.2" [label="VI.def.2", fillcolor="#224422", tooltip="VI.def.2: [2. Reciprocally related figures. See note.]", URL="../../../../volume_II/book_vi/definitions/index.html#vi-def-2", target="_top"]; "VI.def.4" [label="VI.def.4", fillcolor="#224422", tooltip="VI.def.4: The height of any figure is the perpendicular drawn from the vertex to the base.", URL="../../../../volume_II/book_vi/definitions/index.html#vi-def-4", target="_top"]; "VI.1" [label="VI.1", fillcolor="#222244", tooltip="VI.1: Triangles and parallelograms which are under the same height are to one another as their bases.", URL="../../../../volume_II/book_vi/propositions/01/index.html", target="_top"]; "VI.2" [label="VI.2", fillcolor="#222244", tooltip="VI.2: If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally; and, if the sides of the triangle be cut…", URL="../../../../volume_II/book_vi/propositions/02/index.html", target="_top"]; "VI.4" [label="VI.4", fillcolor="#222244", tooltip="VI.4: In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles.", URL="../../../../volume_II/book_vi/propositions/04/index.html", target="_top"]; "VI.6" [label="VI.6", fillcolor="#222244", tooltip="VI.6: If two triangles have one angle equal to one angle and the sides about the equal angles proportional, the triangles will be equiangular and will have those angles equal which the…", URL="../../../../volume_II/book_vi/propositions/06/index.html", target="_top"]; "VI.11" [label="VI.11", fillcolor="#222244", tooltip="VI.11: To two given straight lines to find a third proportional.", URL="../../../../volume_II/book_vi/propositions/11/index.html", target="_top"]; "VI.12" [label="VI.12", fillcolor="#222244", tooltip="VI.12: To three given straight lines to find a fourth proportional.", URL="../../../../volume_II/book_vi/propositions/12/index.html", target="_top"]; "VI.14" [label="VI.14", fillcolor="#222244", tooltip="VI.14: In equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; and equiangular parallelograms in which the sides about the equal angles…", URL="../../../../volume_II/book_vi/propositions/14/index.html", target="_top"]; "VI.15" [label="VI.15", fillcolor="#222244", tooltip="VI.15: In equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally proportional; and those triangles which have one angle equal to one…", URL="../../../../volume_II/book_vi/propositions/15/index.html", target="_top"]; "VI.18" [label="VI.18", fillcolor="#222244", tooltip="VI.18: On a given straight line to describe a rectilinear figure similar and similarly situated to a given rectilinear figure.", URL="../../../../volume_II/book_vi/propositions/18/index.html", target="_top"]; "VI.19" [label="VI.19", fillcolor="#222244", tooltip="VI.19: Similar triangles are to one another in the duplicate ratio of the corresponding sides.", URL="../../../../volume_II/book_vi/propositions/19/index.html", target="_top"]; "VI.20" [label="VI.20", fillcolor="#222244", tooltip="VI.20: Similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio…", URL="../../../../volume_II/book_vi/propositions/20/index.html", target="_top"]; "VI.22" [label="VI.22", fillcolor="#222244", tooltip="VI.22: If four straight lines be proportional, the rectilineal figures similar and similarly described upon them will also be proportional; and, if the rectilineal figures similar and…", URL="../../../../volume_II/book_vi/propositions/22/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/book_vii/definitions/index.html#vii-def-16", 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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/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="../../../../volume_II/book_vii/propositions/33/index.html", target="_top"]; "VII.34" [label="VII.34", fillcolor="#222244", tooltip="VII.34: Given two numbers, to find the least number which they measure.", URL="../../../../volume_II/book_vii/propositions/34/index.html", target="_top"]; "VII.35" [label="VII.35", fillcolor="#222244", tooltip="VII.35: If two numbers measure any number, the least number measured by them will also measure the same.", URL="../../../../volume_II/book_vii/propositions/35/index.html", target="_top"]; "VIII.def.20" [label="VIII.def.20", fillcolor="#224422", tooltip="VIII.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="../../../../volume_II/book_vii/definitions/index.html#vii-def-20", 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="../../../../volume_II/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="../../../../volume_II/book_viii/propositions/02/index.html", target="_top"]; "VIII.4" [label="VIII.4", fillcolor="#222244", tooltip="VIII.4: Given as many ratios as we please in least numbers, to find numbers in continued proportion which are the least in the given ratios.", URL="../../../../volume_II/book_viii/propositions/04/index.html", target="_top"]; "VIII.11" [label="VIII.11", fillcolor="#222244", 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="../../../../volume_II/book_viii/propositions/11/index.html", target="_top"]; "VIII.18" [label="VIII.18", fillcolor="#222244", 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="../../../../volume_II/book_viii/propositions/18/index.html", target="_top"]; "VIII.26" [label="VIII.26", fillcolor="#222244", tooltip="VIII.26: Similar plane numbers have to one another the ratio which a square number has to a square number.", URL="../../../../volume_II/book_viii/propositions/26/index.html", target="_top"]; "X.def.1" [label="X.def.1", fillcolor="#224422", tooltip="X.def.1: Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.", URL="../../definitions/index.html#x-def-1", target="_top"]; "X.def.2" [label="X.def.2", fillcolor="#224422", tooltip="X.def.2: Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when the squares on them cannot possibly have any…", URL="../../definitions/index.html#x-def-2", target="_top"]; "X.def.3" [label="X.def.3", fillcolor="#224422", tooltip="X.def.3: With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others…", URL="../../definitions/index.html#x-def-3", target="_top"]; "X.def.4" [label="X.def.4", fillcolor="#224422", tooltip="X.def.4: And let the square on the assigned straight line be called rational and those areas which are commensurable with it rational, but those which are incommensurable with it…", URL="../../definitions/index.html#x-def-4", target="_top"]; "X.def.ii.6" [label="X.def.ii.6", fillcolor="#224422", tooltip="X.def.ii.6: and if neither, a sixth binomial.", URL="../../definitions_ii/index.html#x-def-ii-6", target="_top"]; "X.5" [label="X.5", fillcolor="#222244", tooltip="X.5: Commensurable magnitudes have to one another the ratio which a number has to a number.", URL="../005/index.html", target="_top"]; "X.6" [label="X.6", fillcolor="#222244", tooltip="X.6: If two magnitudes have to one another the ratio which a number has to a number, the magnitudes will be commensurable.", URL="../006/index.html", target="_top"]; "X.7" [label="X.7", fillcolor="#222244", tooltip="X.7: Incommensurable magnitudes have not to one another the ratio which a number has to a number.", URL="../007/index.html", target="_top"]; "X.8" [label="X.8", fillcolor="#222244", tooltip="X.8: If two magnitudes have not to one another the ratio which a number has to a number, the magnitudes will be incommensurable.", URL="../008/index.html", target="_top"]; "X.9" [label="X.9", fillcolor="#222244", tooltip="X.9: The squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio…", URL="../009/index.html", target="_top"]; "X.11" [label="X.11", fillcolor="#222244", tooltip="X.11: If four magnitudes be proportional, and the first be commensurable with the second, the third will also be commensurable with the fourth ; and, if the first be incommensurable…", URL="../011/index.html", target="_top"]; "X.12" [label="X.12", fillcolor="#222244", tooltip="X.12: Magnitudes commensurable with the same magnitude are commensurable with one another also.", URL="../012/index.html", target="_top"]; "X.13" [label="X.13", fillcolor="#222244", tooltip="X.13: If two magnitudes be commensurable, and the one of them be incommensurable with any magnitude, the remaining one will also be incommensurable with the same.", URL="../013/index.html", target="_top"]; "X.14" [label="X.14", fillcolor="#222244", tooltip="X.14: If four straight lines be proportional, and the square on the first be greater than the square on the second by the square on a straight line commensurable with the first, the…", URL="../014/index.html", target="_top"]; "X.15" [label="X.15", fillcolor="#222244", tooltip="X.15: If two commensurable magnitudes be added together, the whole will also be commensurable with each of them; and, if the whole be commensurable with one of them, the original…", URL="../015/index.html", target="_top"]; "X.16" [label="X.16", fillcolor="#222244", tooltip="X.16: If two incommensurable magnitudes be added together, the whole will also be incommensurable with each of them; and, if the whole be incommensurable with one of them, the original…", URL="../016/index.html", target="_top"]; "X.17" [label="X.17", fillcolor="#222244", tooltip="X.17: If there be two unequal straight lines, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure…", URL="../017/index.html", target="_top"]; "X.18" [label="X.18", fillcolor="#222244", tooltip="X.18: If there be two unequal straight lines, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure…", URL="../018/index.html", target="_top"]; "X.21" [label="X.21", fillcolor="#222244", tooltip="X.21: The rectangle contained by rational straight lines commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called…", URL="../021/index.html", target="_top"]; "X.22" [label="X.22", fillcolor="#222244", tooltip="X.22: The square on a medial straight line, if applied to a rational straight line, produces as breadth a straight line rational and incommensurable in length with that to which it is…", URL="../022/index.html", target="_top"]; "X.23" [label="X.23", fillcolor="#222244", tooltip="X.23: A straight line commensurable with a medial straight line is medial.", URL="../023/index.html", target="_top"]; "X.29" [label="X.29", fillcolor="#222244", tooltip="X.29: To find two rational straight lines commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a straight line…", URL="../029/index.html", target="_top"]; "X.30" [label="X.30", fillcolor="#222244", tooltip="X.30: To find two rational straight lines commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a straight line…", URL="../030/index.html", target="_top"]; "X.32" [label="X.32", fillcolor="#222244", tooltip="X.32: To find two medial straight lines commensurable in square only, containing a medial rectangle, and such that the square on the greater is greater than the square on the less by…", URL="../032/index.html", target="_top"]; "X.35" [label="X.35", fillcolor="#222244", tooltip="X.35: To find two straight lines incommensurable in square which make the sum of the squares on them medial and the rectangle contained by them medial and moreover incommensurable with…", URL="../035/index.html", target="_top"]; "X.36" [label="X.36", fillcolor="#222244", tooltip="X.36: If two rational straight lines commensurable in square only be added together, the whole is irrational; and let it be called binomial.", URL="../036/index.html", target="_top"]; "X.41" [label="X.41", fillcolor="#222244", tooltip="X.41: If two straight lines incommensurable in square which make the sum of the squares on them medial, and the rectangle contained by them medial and also incommensurable with the sum…", URL="../041/index.html", target="_top"]; 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"I.def.19" -> "I.def.4"; "I.def.23" -> "I.def.4"; "I.post.1" -> "I.def.4"; "I.post.2" -> "I.def.4"; "I.post.5" -> "I.def.4"; "I.1" -> "I.def.4"; "I.2" -> "I.def.4"; "I.3" -> "I.def.4"; "I.4" -> "I.def.4"; "I.5" -> "I.def.4"; "I.7" -> "I.def.4"; "I.8" -> "I.def.4"; "I.10" -> "I.def.4"; "I.11" -> "I.def.4"; "I.13" -> "I.def.4"; "I.14" -> "I.def.4"; "I.15" -> "I.def.4"; "I.22" -> "I.def.4"; "I.23" -> "I.def.4"; "I.27" -> "I.def.4"; "I.28" -> "I.def.4"; "I.29" -> "I.def.4"; "I.31" -> "I.def.4"; "I.33" -> "I.def.4"; "I.46" -> "I.def.4"; "II.def.1" -> "I.def.4"; "II.4" -> "I.def.4"; "II.5" -> "I.def.4"; "III.def.8" -> "I.def.4"; "III.def.10" -> "I.def.4"; "IV.def.7" -> "I.def.4"; "IV.1" -> "I.def.4"; "VI.2" -> "I.def.4"; "VI.11" -> "I.def.4"; "VI.12" -> "I.def.4"; "VI.18" -> "I.def.4"; "VI.22" -> "I.def.4"; "X.def.2" -> "I.def.4"; "X.def.3" -> "I.def.4"; "X.def.4" -> "I.def.4"; "X.9" -> "I.def.4"; "X.14" -> "I.def.4"; "X.16" -> "I.def.4"; "X.17" -> "I.def.4"; "X.18" -> "I.def.4"; "X.21" -> "I.def.4"; "X.22" -> "I.def.4"; "X.23" -> "I.def.4"; "X.29" -> "I.def.4"; "X.30" -> "I.def.4"; "X.32" -> "I.def.4"; "X.35" -> "I.def.4"; "X.36" -> "I.def.4"; "X.41" -> "I.def.4"; "X.65" -> "I.def.4"; "I.def.6" -> "I.def.5"; "I.def.7" -> "I.def.5"; "I.def.15" -> "I.def.6"; "I.def.8" -> "I.def.7"; "I.def.15" -> "I.def.7"; "I.def.23" -> "I.def.7"; "VIII.18" -> "I.def.7"; "I.def.9" -> "I.def.8"; "I.def.10" -> "I.def.8"; "I.def.11" -> "I.def.8"; "I.def.12" -> "I.def.8"; "I.def.21" -> "I.def.8"; "I.def.22" -> "I.def.8"; "I.post.5" -> "I.def.8"; "I.4" -> "I.def.8"; "I.5" -> "I.def.8"; "I.6" -> "I.def.8"; "I.8" -> "I.def.8"; "I.13" -> "I.def.8"; "I.14" -> "I.def.8"; "I.15" -> "I.def.8"; "I.16" -> "I.def.8"; "I.17" -> "I.def.8"; "I.18" -> "I.def.8"; "I.19" -> "I.def.8"; "I.26" -> "I.def.8"; "I.27" -> "I.def.8"; "I.28" -> "I.def.8"; "I.29" -> "I.def.8"; "I.32" -> "I.def.8"; "I.34" -> "I.def.8"; "III.def.8" -> "I.def.8"; "III.def.10" -> "I.def.8"; "III.20" -> "I.def.8"; "III.21" -> "I.def.8"; "III.22" -> "I.def.8"; "III.31" -> "I.def.8"; "VI.def.1" -> "I.def.8"; "VI.4" -> "I.def.8"; "VI.6" -> "I.def.8"; "VI.14" -> "I.def.8"; "VI.15" -> "I.def.8"; "I.9" -> "I.def.9"; "I.23" -> "I.def.9"; "I.def.11" -> "I.def.10"; "I.def.12" -> "I.def.10"; "I.def.21" -> "I.def.10"; "I.post.4" -> "I.def.10"; "I.post.5" -> "I.def.10"; "I.11" -> "I.def.10"; "I.13" -> "I.def.10"; "I.14" -> "I.def.10"; "I.17" -> "I.def.10"; "I.28" -> "I.def.10"; "I.29" -> "I.def.10"; "I.32" -> "I.def.10"; "I.47" -> "I.def.10"; "II.def.1" -> "I.def.10"; "III.22" -> "I.def.10"; "III.31" -> "I.def.10"; "VI.def.4" -> "I.def.10"; "I.def.21" -> "I.def.11"; "I.def.21" -> "I.def.12"; "I.def.14" -> "I.def.13"; "I.def.15" -> "I.def.14"; "I.def.18" -> "I.def.14"; "I.4" -> "I.def.14"; "I.6" -> "I.def.14"; "III.def.10" -> "I.def.14"; "V.9" -> "I.def.14"; "V.10" -> "I.def.14"; "V.11" -> "I.def.14"; "VI.def.4" -> "I.def.14"; "VI.18" -> "I.def.14"; "X.16" -> "I.def.14"; "X.17" -> "I.def.14"; "X.18" -> "I.def.14"; "I.def.16" -> "I.def.15"; "I.def.17" -> "I.def.15"; "I.def.18" -> "I.def.15"; "I.post.3" -> "I.def.15"; "I.1" -> "I.def.15"; "I.3" -> "I.def.15"; "III.def.10" -> "I.def.15"; "III.20" -> "I.def.15"; "III.21" -> "I.def.15"; "III.22" -> "I.def.15"; "III.31" -> "I.def.15"; "IV.def.7" -> "I.def.15"; "IV.1" -> "I.def.15"; "I.def.17" -> "I.def.16"; "I.def.18" -> "I.def.16"; "I.post.3" -> "I.def.16"; "III.def.10" -> "I.def.16"; "I.def.18" -> "I.def.17"; "I.34" -> "I.def.17"; "I.43" -> "I.def.17"; "II.def.2" -> "I.def.17"; "III.31" -> "I.def.17"; "IV.1" -> "I.def.17"; "III.31" -> "I.def.18"; "I.def.20" -> "I.def.19"; "I.def.21" -> "I.def.19"; "I.def.22" -> "I.def.19"; "I.4" -> "I.def.19"; "I.6" -> "I.def.19"; "I.8" -> "I.def.19"; "I.16" -> "I.def.19"; "I.17" -> "I.def.19"; "I.18" -> "I.def.19"; "I.19" -> "I.def.19"; "I.20" -> "I.def.19"; "I.22" -> "I.def.19"; "I.26" -> "I.def.19"; "I.32" -> "I.def.19"; "I.37" -> "I.def.19"; "I.38" -> "I.def.19"; "I.39" -> "I.def.19"; "I.41" -> "I.def.19"; "III.22" -> "I.def.19"; "VI.1" -> "I.def.19"; "VI.2" -> "I.def.19"; "VI.4" -> "I.def.19"; "VI.6" -> "I.def.19"; "VI.15" -> "I.def.19"; "VI.20" -> "I.def.19"; "VI.22" -> "I.def.19"; "X.def.4" -> "I.def.19"; "I.def.22" -> "I.def.20"; "I.1" -> "I.def.20"; "I.5" -> "I.def.20"; "I.47" -> "I.def.21"; "I.35" -> "I.def.22"; "I.46" -> "I.def.22"; "I.47" -> "I.def.22"; "II.4" -> "I.def.22"; "II.5" -> "I.def.22"; "VIII.2" -> "I.def.22"; "VIII.11" -> "I.def.22"; "X.def.2" -> "I.def.22"; "X.def.4" -> "I.def.22"; "X.9" -> "I.def.22"; "X.14" -> "I.def.22"; "X.16" -> "I.def.22"; "X.17" -> "I.def.22"; "X.18" -> "I.def.22"; "X.21" -> "I.def.22"; "X.22" -> "I.def.22"; "X.29" -> "I.def.22"; "X.30" -> "I.def.22"; "X.32" -> "I.def.22"; "X.35" -> "I.def.22"; "X.41" -> "I.def.22"; "X.65" -> "I.def.22"; "I.27" -> "I.def.23"; "I.28" -> "I.def.23"; "I.29" -> "I.def.23"; "I.31" -> "I.def.23"; "I.33" -> "I.def.23"; "I.35" -> "I.def.23"; "I.36" -> "I.def.23"; "I.37" -> "I.def.23"; "I.38" -> "I.def.23"; "I.39" -> "I.def.23"; "I.41" -> "I.def.23"; "VI.2" -> "I.def.23"; "I.1" -> "I.post.1"; "I.2" -> "I.post.1"; "I.5" -> "I.post.1"; "I.7" -> "I.post.1"; "I.16" -> "I.post.1"; "I.2" -> "I.post.2"; "I.5" -> "I.post.2"; "I.16" -> "I.post.2"; "I.17" -> "I.post.2"; "I.1" -> "I.post.3"; "I.2" -> "I.post.3"; "I.3" -> "I.post.3"; "I.14" -> "I.post.4"; "I.15" -> "I.post.4"; "I.29" -> "I.post.5"; "VI.4" -> "I.post.5"; "I.1" -> "I.cn.1"; "I.2" -> "I.cn.1"; "I.3" -> "I.cn.1"; "I.13" -> "I.cn.1"; "I.14" -> "I.cn.1"; "I.15" -> "I.cn.1"; "I.29" -> "I.cn.1"; "I.35" -> "I.cn.1"; "I.36" -> "I.cn.1"; "I.39" -> "I.cn.1"; "I.13" -> "I.cn.2"; "I.29" -> "I.cn.2"; "I.34" -> "I.cn.2"; "I.35" -> "I.cn.2"; "I.43" -> "I.cn.2"; "I.47" -> "I.cn.2"; "I.2" -> "I.cn.3"; "I.14" -> "I.cn.3"; "I.15" -> "I.cn.3"; "I.35" -> "I.cn.3"; "I.43" -> "I.cn.3"; "I.4" -> "I.cn.4"; "I.16" -> "I.cn.5"; "I.20" -> "I.cn.5"; "I.2" -> "I.1"; "I.10" -> "I.1"; "I.11" -> "I.1"; "I.3" -> "I.2"; "I.5" -> "I.3"; "I.9" -> "I.3"; "I.11" -> "I.3"; "I.16" -> "I.3"; "I.18" -> "I.3"; "I.22" -> "I.3"; "VI.11" -> "I.3"; "I.5" -> "I.4"; "I.10" -> "I.4"; "I.16" -> "I.4"; "I.26" -> "I.4"; "I.33" -> "I.4"; "I.34" -> "I.4"; "I.35" -> "I.4"; "I.47" -> "I.4"; "VI.6" -> "I.4"; "I.7" -> "I.5"; "I.19" -> "I.5"; "I.20" -> "I.5"; "II.4" -> "I.5"; "III.20" -> "I.5"; "III.31" -> "I.5"; "II.4" -> "I.6"; "I.8" -> "I.7"; "I.9" -> "I.8"; "I.11" -> "I.8"; "I.23" -> "I.8"; "I.10" -> "I.9"; "I.16" -> "I.10"; "I.13" -> "I.11"; "I.46" -> "I.11"; "I.14" -> "I.13"; "I.15" -> "I.13"; "I.17" -> "I.13"; "I.28" -> "I.13"; "I.29" -> "I.13"; "I.32" -> "I.13"; "I.47" -> "I.14"; "VI.14" -> "I.14"; "VI.15" -> "I.14"; "I.16" -> "I.15"; "I.28" -> "I.15"; "I.29" -> "I.15"; "I.18" -> "I.16"; "I.26" -> "I.16"; "I.27" -> "I.16"; "III.31" -> "I.17"; "VI.4" -> "I.17"; "I.19" -> "I.18"; "I.20" -> "I.19"; "I.22" -> "I.20"; "I.23" -> "I.22"; "I.31" -> "I.23"; "VI.6" -> "I.23"; "VI.18" -> "I.23"; "I.34" -> "I.26"; "I.28" -> "I.27"; "I.31" -> "I.27"; "I.33" -> "I.27"; "VI.4" -> "I.28"; "I.32" -> "I.29"; "I.33" -> "I.29"; "I.34" -> "I.29"; "I.35" -> "I.29"; "I.46" -> "I.29"; "II.4" -> "I.29"; "I.32" -> "I.31"; "I.37" -> "I.31"; "I.38" -> "I.31"; "I.39" -> "I.31"; "I.46" -> "I.31"; "II.4" -> "I.31"; "II.5" -> "I.31"; "VI.11" -> "I.31"; "VI.12" -> "I.31"; "III.20" -> "I.32"; "III.22" -> "I.32"; "III.31" -> "I.32"; "VI.6" -> "I.32"; "VI.18" -> "I.32"; "VI.20" -> "I.32"; "I.36" -> "I.33"; "I.35" -> "I.34"; "I.36" -> "I.34"; "I.37" -> "I.34"; "I.38" -> "I.34"; "I.41" -> "I.34"; "I.43" -> "I.34"; "I.46" -> "I.34"; "II.4" -> "I.34"; "VI.4" -> "I.34"; "I.36" -> "I.35"; "I.37" -> "I.35"; "I.38" -> "I.36"; "II.5" -> "I.36"; "I.39" -> "I.37"; "I.41" -> "I.37"; "VI.1" -> "I.38"; "VI.2" -> "I.38"; "VI.2" -> "I.39"; "I.47" -> "I.41"; "VI.1" -> "I.41"; "II.5" -> "I.43"; "I.47" -> "I.46"; "II.4" -> "I.46"; "II.5" -> "I.46"; "X.13" -> "I.47"; "X.29" -> "I.47"; "X.30" -> "I.47"; "X.35" -> "I.47"; "II.4" -> "II.def.1"; "II.5" -> "II.def.1"; "X.16" -> "II.def.1"; "X.21" -> "II.def.1"; "X.32" -> "II.def.1"; "X.35" -> "II.def.1"; "X.41" -> "II.def.1"; "II.5" -> "II.def.2"; "X.36" -> "II.4"; "X.41" -> "II.4"; "X.17" -> "II.5"; "III.31" -> "III.def.8"; "III.20" -> "III.def.10"; "III.21" -> "III.20"; "III.22" -> "III.21"; "III.31" -> "III.22"; "X.13" -> "III.31"; "X.29" -> "III.31"; "X.30" -> "III.31"; "X.35" -> "III.31"; "IV.1" -> "IV.def.7"; "X.13" -> "IV.1"; "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"; "VII.34" -> "V.def.1"; "VII.35" -> "V.def.1"; "X.def.1" -> "V.def.1"; "X.def.2" -> "V.def.1"; "V.def.5" -> "V.def.2"; "V.def.7" -> "V.def.2"; "V.1" -> "V.def.2"; "V.2" -> "V.def.2"; "V.3" -> "V.def.2"; "V.4" -> "V.def.2"; "V.15" -> "V.def.2"; "VII.5" -> "V.def.2"; "V.def.17" -> "V.def.3"; "V.10" -> "V.def.3"; "V.11" -> "V.def.3"; "VI.20" -> "V.def.3"; "VII.33" -> "V.def.3"; "VIII.2" -> "V.def.3"; "VIII.4" -> "V.def.3"; "VIII.11" -> "V.def.3"; "VIII.18" -> "V.def.3"; "VIII.26" -> "V.def.3"; "X.5" -> "V.def.3"; "X.6" -> "V.def.3"; "X.7" -> "V.def.3"; "X.8" -> "V.def.3"; "X.9" -> "V.def.3"; "V.8" -> "V.def.4"; "V.10" -> "V.def.4"; "V.def.6" -> "V.def.5"; "V.4" -> "V.def.5"; "V.7" -> "V.def.5"; "V.9" -> "V.def.5"; "V.11" -> "V.def.5"; "V.12" -> "V.def.5"; "V.13" -> "V.def.5"; "V.14" -> "V.def.5"; "V.15" -> "V.def.5"; "V.16" -> "V.def.5"; "V.20" -> "V.def.5"; "V.22" -> "V.def.5"; "VI.1" -> "V.def.5"; "VI.20" -> "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"; "V.def.9" -> "V.def.6"; "V.7" -> "V.def.6"; "V.12" -> "V.def.6"; "V.16" -> "V.def.6"; "V.17" -> "V.def.6"; "VI.def.1" -> "V.def.6"; "VI.4" -> "V.def.6"; "VI.6" -> "V.def.6"; "VI.11" -> "V.def.6"; "VI.12" -> "V.def.6"; "VI.22" -> "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.4" -> "V.def.6"; "VIII.11" -> "V.def.6"; "VIII.18" -> "V.def.6"; "X.11" -> "V.def.6"; "X.14" -> "V.def.6"; "V.8" -> "V.def.7"; "V.10" -> "V.def.7"; "V.13" -> "V.def.7"; "VI.19" -> "V.def.9"; "VIII.11" -> "V.def.9"; "VIII.18" -> "V.def.9"; "V.def.12" -> "V.def.11"; "V.def.13" -> "V.def.11"; "V.def.14" -> "V.def.11"; "V.def.15" -> "V.def.11"; "V.def.16" -> "V.def.11"; "V.12" -> "V.def.11"; "VI.19" -> "V.def.11"; "VII.12" -> "V.def.11"; "V.16" -> "V.def.12"; "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"; "V.17" -> "V.def.14"; "X.36" -> "V.def.14"; "V.17" -> "V.def.15"; "X.17" -> "V.def.15"; "V.19" -> "V.def.16"; "V.3" -> "V.def.17"; "V.20" -> "V.def.17"; "V.22" -> "V.def.17"; "VII.14" -> "V.def.17"; "V.8" -> "V.1"; "V.12" -> "V.1"; "V.17" -> "V.1"; "V.3" -> "V.2"; "V.17" -> "V.2"; "V.4" -> "V.3"; "V.22" -> "V.4"; "V.10" -> "V.7"; "V.15" -> "V.7"; "VI.2" -> "V.7"; "VI.14" -> "V.7"; "VI.15" -> "V.7"; "VII.19" -> "V.7"; "X.5" -> "V.7"; "X.6" -> "V.7"; "V.9" -> "V.8"; "V.10" -> "V.8"; "V.14" -> "V.8"; "V.20" -> "V.8"; "VI.2" -> "V.9"; "VI.6" -> "V.9"; "VI.14" -> "V.9"; "VI.15" -> "V.9"; "VI.22" -> "V.9"; "VII.19" -> "V.9"; "X.6" -> "V.9"; "V.14" -> "V.10"; "V.20" -> "V.10"; "V.16" -> "V.11"; "V.19" -> "V.11"; "VI.1" -> "V.11"; "VI.2" -> "V.11"; "VI.6" -> "V.11"; "VI.14" -> "V.11"; "VI.15" -> "V.11"; "VI.19" -> "V.11"; "VI.22" -> "V.11"; "X.6" -> "V.11"; "X.12" -> "V.11"; "V.15" -> "V.12"; "VI.20" -> "V.12"; "V.14" -> "V.13"; "V.20" -> "V.13"; "V.16" -> "V.14"; "V.16" -> "V.15"; "VI.1" -> "V.15"; "V.19" -> "V.16"; "VI.4" -> "V.16"; "VI.19" -> "V.16"; "V.19" -> "V.17"; "X.14" -> "V.17"; "X.29" -> "V.19"; "X.30" -> "V.19"; "V.22" -> "V.20"; "VI.4" -> "V.22"; "VI.20" -> "V.22"; "VI.22" -> "V.22"; "X.5" -> "V.22"; "X.6" -> "V.22"; "X.12" -> "V.22"; "X.14" -> "V.22"; "VI.18" -> "VI.def.1"; "VI.19" -> "VI.def.1"; "VI.20" -> "VI.def.1"; "VI.22" -> "VI.def.1"; "VI.14" -> "VI.def.2"; "VI.15" -> "VI.def.2"; "VI.1" -> "VI.def.4"; "VI.2" -> "VI.1"; "VI.14" -> "VI.1"; "VI.15" -> "VI.1"; "VI.19" -> "VI.1"; "VI.20" -> "VI.1"; "X.21" -> "VI.1"; "X.23" -> "VI.1"; "X.41" -> "VI.1"; "X.65" -> "VI.1"; "VI.4" -> "VI.2"; "VI.11" -> "VI.2"; "VI.12" -> "VI.2"; "VI.6" -> "VI.4"; "VI.18" -> "VI.4"; "VI.20" -> "VI.4"; "VI.20" -> "VI.6"; "VI.19" -> "VI.11"; "VI.22" -> "VI.11"; "VI.22" -> "VI.12"; "X.22" -> "VI.14"; "VI.19" -> "VI.15"; "VI.22" -> "VI.18"; "VI.20" -> "VI.19"; "VI.22" -> "VI.19"; "X.6" -> "VI.19"; "X.9" -> "VI.20"; "X.14" -> "VI.22"; "X.22" -> "VI.22"; "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.def.16" -> "VII.def.2"; "VII.def.18" -> "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"; "VII.34" -> "VII.def.2"; "VII.35" -> "VII.def.2"; "VIII.1" -> "VII.def.2"; "VIII.2" -> "VII.def.2"; "VIII.4" -> "VII.def.2"; "VIII.11" -> "VII.def.2"; "VIII.18" -> "VII.def.2"; "X.5" -> "VII.def.2"; "X.6" -> "VII.def.2"; "X.7" -> "VII.def.2"; "X.8" -> "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"; "X.17" -> "VII.def.3"; "X.18" -> "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"; "VII.34" -> "VII.def.11"; "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"; "VII.def.16" -> "VII.def.15"; "VII.def.18" -> "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"; "VII.34" -> "VII.def.15"; "VIII.18" -> "VII.def.16"; "VIII.11" -> "VII.def.18"; "VIII.26" -> "VII.def.18"; "X.9" -> "VII.def.18"; "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.4" -> "VII.def.20"; "X.5" -> "VII.def.20"; "X.6" -> "VII.def.20"; "VIII.18" -> "VII.def.21"; "VIII.26" -> "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.4" -> "VII.13"; "VIII.18" -> "VII.13"; "VIII.1" -> "VII.14"; "VII.16" -> "VII.15"; "VII.18" -> "VII.16"; "VII.21" -> "VII.16"; "VII.24" -> "VII.16"; "VII.33" -> "VII.16"; "VII.34" -> "VII.16"; "VII.18" -> "VII.17"; "VII.19" -> "VII.17"; "VII.22" -> "VII.17"; "VII.34" -> "VII.17"; "VIII.2" -> "VII.17"; "VIII.11" -> "VII.17"; "VIII.18" -> "VII.17"; "VII.19" -> "VII.18"; "VIII.2" -> "VII.18"; "VIII.11" -> "VII.18"; "VII.24" -> "VII.19"; "VII.33" -> "VII.19"; "VII.34" -> "VII.19"; "VII.21" -> "VII.20"; "VII.24" -> "VII.20"; "VII.34" -> "VII.20"; "VIII.1" -> "VII.20"; "VIII.4" -> "VII.20"; "VII.24" -> "VII.21"; "VII.33" -> "VII.21"; "VII.34" -> "VII.21"; "VIII.1" -> "VII.21"; "VIII.2" -> "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"; "VII.34" -> "VII.33"; "VIII.26" -> "VII.33"; "VIII.4" -> "VII.34"; "VIII.4" -> "VII.35"; "VIII.4" -> "VIII.def.20"; "VIII.2" -> "VIII.1"; "VIII.26" -> "VIII.2"; "X.12" -> "VIII.4"; "X.9" -> "VIII.11"; "VIII.26" -> "VIII.18"; "X.9" -> "VIII.26"; "X.def.3" -> "X.def.1"; "X.def.4" -> "X.def.1"; "X.5" -> "X.def.1"; "X.6" -> "X.def.1"; "X.7" -> "X.def.1"; "X.8" -> "X.def.1"; "X.9" -> "X.def.1"; "X.11" -> "X.def.1"; "X.12" -> "X.def.1"; "X.13" -> "X.def.1"; "X.14" -> "X.def.1"; "X.15" -> "X.def.1"; "X.16" -> "X.def.1"; "X.17" -> "X.def.1"; "X.18" -> "X.def.1"; "X.22" -> "X.def.1"; "X.23" -> "X.def.1"; "X.29" -> "X.def.1"; "X.30" -> "X.def.1"; "X.32" -> "X.def.1"; "X.35" -> "X.def.1"; "X.41" -> "X.def.1"; "X.9" -> "X.def.2"; "X.21" -> "X.def.2"; "X.29" -> "X.def.2"; "X.30" -> "X.def.2"; "X.32" -> "X.def.2"; "X.35" -> "X.def.2"; "X.36" -> "X.def.2"; "X.41" -> "X.def.2"; "X.21" -> "X.def.3"; "X.22" -> "X.def.3"; "X.23" -> "X.def.3"; "X.29" -> "X.def.3"; "X.30" -> "X.def.3"; "X.36" -> "X.def.3"; "X.41" -> "X.def.3"; "X.65" -> "X.def.3"; "X.21" -> "X.def.4"; "X.22" -> "X.def.4"; "X.29" -> "X.def.4"; "X.36" -> "X.def.4"; "X.41" -> "X.def.4"; "X.65" -> "X.def.ii.6"; "X.8" -> "X.5"; "X.9" -> "X.5"; "X.11" -> "X.5"; "X.12" -> "X.5"; "X.7" -> "X.6"; "X.9" -> "X.6"; "X.11" -> "X.6"; "X.12" -> "X.6"; "X.17" -> "X.6"; "X.18" -> "X.6"; "X.29" -> "X.6"; "X.30" -> "X.6"; "X.36" -> "X.6"; "X.11" -> "X.7"; "X.11" -> "X.8"; "X.29" -> "X.9"; "X.30" -> "X.9"; "X.14" -> "X.11"; "X.21" -> "X.11"; "X.22" -> "X.11"; "X.23" -> "X.11"; "X.32" -> "X.11"; "X.35" -> "X.11"; "X.36" -> "X.11"; "X.41" -> "X.11"; "X.65" -> "X.11"; "X.13" -> "X.12"; "X.17" -> "X.12"; "X.18" -> "X.13"; "X.22" -> "X.13"; "X.23" -> "X.13"; "X.35" -> "X.13"; "X.36" -> "X.13"; "X.32" -> "X.14"; "X.17" -> "X.15"; "X.36" -> "X.15"; "X.18" -> "X.16"; "X.36" -> "X.16"; "X.18" -> "X.17"; "X.35" -> "X.18"; "X.22" -> "X.21"; "X.23" -> "X.21"; "X.32" -> "X.21"; "X.23" -> "X.22"; "X.41" -> "X.22"; "X.65" -> "X.22"; "X.32" -> "X.23"; "X.32" -> "X.29"; "X.32" -> "X.30"; "X.35" -> "X.32"; "X.41" -> "X.35"; "X.41" -> "X.36"; "X.65" -> "X.36"; "X.65" -> "X.41"; }

See Commentary on Proposition X.65.