PROPOSITION 9.

In equal pyramids which have triangular bases the bases are reciprocally proportional to the heights; and those pyramids in which the bases are reciprocally proportional to the heights are equal.

For let there be equal pyramids which have the triangular bases ABC, DEF and vertices the points G, H; I say that in the pyramids ABCG, DEFH the bases are reciprocally proportional to the heights, that is, as the base ABC is to the base DEF, so is the height of the pyramid DEFH to the height of the pyramid ABCG.

G L A M B C H O R P E D F Q

For let the parallelepipedal solids BGML, EHQP be completed.

Now, since the pyramid ABCG is equal to the pyramid DEFH, and the solid BGML is six times the pyramid ABCG, and the solid EHQP six times the pyramid DEFH,
therefore the solid BGML is equal to the solid EHQP.
But in equal parallelepipedal solids the bases are reciprocally proportional to the heights; [XI. 34]
therefore, as the base BM is to the base EQ, so is the height of the solid EHQP to the height of the solid BGML.
But, as the base BM is to EQ, so is the triangle ABC to the triangle DEF. [I. 34]
Therefore also, as the triangle ABC is to the triangle DEF, so is the height of the solid EHQP to the height of the solid BGML. [V. 11]
But the height of the solid EHQP is the same with the height of the pyramid DEFH, and the height of the solid BGML is the same with the height of the pyramid ABCG,
therefore, as the base ABC is to the base DEF, so is the height of the pyramid DEFH to the height of the pyramid ABCG.
Therefore in the pyramids ABCG, DEFH the bases are reciprocally proportional to the heights.

Next, in the pyramids ABCG, DEFH let the bases be reciprocally proportional to the heights ; that is, as the base ABC is to the base DEF, so let the height of the pyramid DEFH be to the height of the pyramid ABCG ; I say that the pyramid ABCG is equal to the pyramid DEFH.

For, with the same construction, since, as the base ABC is to the base DEF, so is the height of the pyramid DEFH to the height of the pyramid ABCG, while, as the base ABC is to the base DEF, so is the parallelogram BM to the parallelogram EQ,
therefore also, as the parallelogram BM is to the parallelogram EQ, so is the height of the pyramid DEFH to the height of the pyramid ABCG. [V. 11]
But the height of the pyramid DEFH is the same with the height of the parallelepiped EHQP, and the height of the pyramid ABCG is the same with the height of the parallelepiped BGML ;
therefore, as the base BM is to the base EQ, so is the height of the parallelepiped EHQP to the height of the parallelepiped BGML.
But those parallelepipedal solids in which the bases are reciprocally proportional to the heights are equal ; [XI. 34]
therefore the parallelepipedal solid BGML is equal to the parallelepipedal solid EHQP.
And the pyramid ABCG is a sixth part of BGML, and the pyramid DEFH a sixth part of the parallelepiped EHQP ;
therefore the pyramid ABCG is equal to the pyramid DEFH.
Therefore etc
Q. E. D.

Dependency Graph

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"I.def.5" [label="I.def.5", fillcolor="#224422", tooltip="I.def.5: A surface is that which has length and breadth only.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-5", target="_top"]; "I.def.6" [label="I.def.6", fillcolor="#224422", tooltip="I.def.6: The extremities of a surface are lines.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-6", 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.9" [label="I.def.9", fillcolor="#224422", tooltip="I.def.9: And when the lines containing the angle are straight, the angle is called rectilineal.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-9", target="_top"]; "I.def.10" [label="I.def.10", fillcolor="#224422", tooltip="I.def.10: When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-10", 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.15" [label="I.def.15", fillcolor="#224422", tooltip="I.def.15: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-15", target="_top"]; "I.def.16" [label="I.def.16", fillcolor="#224422", tooltip="I.def.16: And the point is called the centre of the circle.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-16", target="_top"]; "I.def.17" [label="I.def.17", fillcolor="#224422", tooltip="I.def.17: A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-17", 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"]; "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.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.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.30" [label="I.30", fillcolor="#222244", tooltip="I.30: Straight lines parallel to the same straight line are also parallel to one another.", URL="../../../../volume_I/book_i/propositions/30/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.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.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.42" [label="I.42", fillcolor="#222244", tooltip="I.42: To construct, in a given rectilineal angle, a parallelogram equal to a given triangle.", URL="../../../../volume_I/book_i/propositions/42/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.44" [label="I.44", fillcolor="#222244", tooltip="I.44: To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle.", URL="../../../../volume_I/book_i/propositions/44/index.html", target="_top"]; "I.45" [label="I.45", fillcolor="#222244", tooltip="I.45: To construct, in a given rectilineal angle, a parallelogram equal to a given rectilineal figure.", URL="../../../../volume_I/book_i/propositions/45/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.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.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.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.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.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.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"]; "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"]; "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.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"]; "XI.def.1" [label="XI.def.1", fillcolor="#224422", tooltip="XI.def.1: A solid is that which has length, breadth, and depth.", URL="../../../book_xi/definitions/index.html#xi-def-1", target="_top"]; "XI.def.2" [label="XI.def.2", fillcolor="#224422", tooltip="XI.def.2: An extremity of a solid is a surface.", URL="../../../book_xi/definitions/index.html#xi-def-2", target="_top"]; "XI.def.3" [label="XI.def.3", fillcolor="#224422", tooltip="XI.def.3: A straight line is at right angles to a plane, when it makes right angles with all the straight lines which meet it and are in the plane.", URL="../../../book_xi/definitions/index.html#xi-def-3", target="_top"]; "XI.def.8" [label="XI.def.8", fillcolor="#224422", tooltip="XI.def.8: Parallel planes are those which do not meet.", URL="../../../book_xi/definitions/index.html#xi-def-8", target="_top"]; "XI.def.10" [label="XI.def.10", fillcolor="#224422", tooltip="XI.def.10: Equal and similar solid figures are those contained by similar planes equal in multitude and in magnitude.", URL="../../../book_xi/definitions/index.html#xi-def-10", target="_top"]; "XI.def.12" [label="XI.def.12", fillcolor="#224422", tooltip="XI.def.12: A pyramid is a solid figure, contained by planes, which is constructed from one plane to one point.", URL="../../../book_xi/definitions/index.html#xi-def-12", target="_top"]; "XI.def.13" [label="XI.def.13", fillcolor="#224422", tooltip="XI.def.13: A prism is a solid figure contained by planes two of which, namely those which are opposite, are equal, similar and parallel, while the rest are parallelograms.", URL="../../../book_xi/definitions/index.html#xi-def-13", target="_top"]; "XI.1" [label="XI.1", fillcolor="#222244", tooltip="XI.1: A part of a straight line cannot be in the plane of reference and a part in a plane more elevated.", URL="../../../book_xi/propositions/01/index.html", target="_top"]; "XI.2" [label="XI.2", fillcolor="#222244", tooltip="XI.2: If two straight lines cut one another, they are in one plane, and every triangle is in one plane.", URL="../../../book_xi/propositions/02/index.html", target="_top"]; "XI.3" [label="XI.3", fillcolor="#222244", tooltip="XI.3: If two planes cut one another, their common section is a straight line.", URL="../../../book_xi/propositions/03/index.html", target="_top"]; "XI.4" [label="XI.4", fillcolor="#222244", tooltip="XI.4: If a straight line be set up at right angles to two straight lines which cut one another, at their common point of section, it will also be at right angles to the plane through…", URL="../../../book_xi/propositions/04/index.html", target="_top"]; "XI.5" [label="XI.5", fillcolor="#222244", tooltip="XI.5: If a straight line be set up at right angles to three straight lines which meet one another, at their common point of section, the three straight lines are in one plane.", URL="../../../book_xi/propositions/05/index.html", target="_top"]; "XI.6" [label="XI.6", fillcolor="#222244", tooltip="XI.6: If two straight lines be at right angles to the same plane, the straight lines will be parallel.", URL="../../../book_xi/propositions/06/index.html", target="_top"]; "XI.7" [label="XI.7", fillcolor="#222244", tooltip="XI.7: If two straight lines be parallel and points be taken at random on each of them, the straight line joining the points is in the same plane with the parallel straight lines.", URL="../../../book_xi/propositions/07/index.html", target="_top"]; "XI.8" [label="XI.8", fillcolor="#222244", tooltip="XI.8: If two straight lines be parallel, and one of them be at right angles to any plane, the remaining one will also be at right angles to the same plane.", URL="../../../book_xi/propositions/08/index.html", target="_top"]; "XI.9" [label="XI.9", fillcolor="#222244", tooltip="XI.9: Straight lines which are parallel to the same straight line and are not in the same plane with it are also parallel to one another.", URL="../../../book_xi/propositions/09/index.html", target="_top"]; "XI.10" [label="XI.10", fillcolor="#222244", tooltip="XI.10: If two straight lines meeting one another be parallel to two straight lines meeting one another not in the same plane, they will contain equal angles.", URL="../../../book_xi/propositions/10/index.html", target="_top"]; "XI.16" [label="XI.16", fillcolor="#222244", tooltip="XI.16: If two parallel planes be cut by any plane, their common sections are parallel.", URL="../../../book_xi/propositions/16/index.html", target="_top"]; "XI.24" [label="XI.24", fillcolor="#222244", tooltip="XI.24: If a solid be contained by parallel planes, the opposite planes in it are equal and parallelogrammic.", URL="../../../book_xi/propositions/24/index.html", target="_top"]; "XI.25" [label="XI.25", fillcolor="#222244", tooltip="XI.25: If a parallelepipedal solid be cut by a plane which is parallel to the opposite planes, then, as the base is to the base, so will the solid be to the solid.", URL="../../../book_xi/propositions/25/index.html", target="_top"]; "XI.29" [label="XI.29", fillcolor="#222244", tooltip="XI.29: Parallelepipedal solids which are on the same base and of the same height, and in which the extremities of the sides which stand up are on the same straight lines, are equal to…", URL="../../../book_xi/propositions/29/index.html", target="_top"]; "XI.30" [label="XI.30", fillcolor="#222244", tooltip="XI.30: Parallelepipedal solids which are on the same base and of the same height, and in which the extremities of the sides which stand up are not on the same straight lines, are equal…", URL="../../../book_xi/propositions/30/index.html", target="_top"]; "XI.31" [label="XI.31", fillcolor="#222244", tooltip="XI.31: Parallelepipedal solids which are on equal bases and of the same height are equal to one another.", URL="../../../book_xi/propositions/31/index.html", target="_top"]; "XI.32" [label="XI.32", fillcolor="#222244", tooltip="XI.32: Parallelepipedal solids which are of the same height are to one another as their bases.", URL="../../../book_xi/propositions/32/index.html", target="_top"]; "XI.34" [label="XI.34", fillcolor="#222244", tooltip="XI.34: In equal parallelepipedal solids the bases are reciprocally proportional to the heights; and those parallelepipedal solids in which the bases are reciprocally proportional to the…", URL="../../../book_xi/propositions/34/index.html", target="_top"]; "XII.9" [label="XII.9", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="XII.9: In equal pyramids which have triangular bases the bases are reciprocally proportional to the heights; and those pyramids in which the bases are reciprocally proportional to the…", URL="index.html", target="_top"]; "I.def.3" -> "I.def.1"; "I.def.4" -> "I.def.1"; "I.def.15" -> "I.def.1"; "I.def.16" -> "I.def.1"; "I.post.1" -> "I.def.1"; "I.2" -> "I.def.1"; "I.7" -> "I.def.1"; "I.11" -> "I.def.1"; "I.14" -> "I.def.1"; "I.23" -> "I.def.1"; "I.31" -> "I.def.1"; "XI.def.12" -> "I.def.1"; "XI.4" -> "I.def.1"; "XI.5" -> "I.def.1"; "XI.7" -> "I.def.1"; "I.def.3" -> "I.def.2"; "I.def.4" -> "I.def.2"; "I.def.6" -> "I.def.2"; "I.def.8" -> "I.def.2"; "I.def.9" -> "I.def.2"; "I.def.4" -> "I.def.3"; "I.def.13" -> "I.def.3"; "I.2" -> "I.def.3"; "I.7" -> "I.def.3"; "I.33" -> "I.def.3"; "XI.def.2" -> "I.def.3"; "XI.29" -> "I.def.3"; "XI.30" -> "I.def.3"; "I.def.7" -> "I.def.4"; "I.def.8" -> "I.def.4"; "I.def.10" -> "I.def.4"; "I.def.15" -> "I.def.4"; "I.def.17" -> "I.def.4"; "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.30" -> "I.def.4"; "I.31" -> "I.def.4"; "I.33" -> "I.def.4"; "I.44" -> "I.def.4"; "XI.def.3" -> "I.def.4"; "XI.1" -> "I.def.4"; "XI.2" -> "I.def.4"; "XI.3" -> "I.def.4"; "XI.4" -> "I.def.4"; "XI.5" -> "I.def.4"; "XI.6" -> "I.def.4"; "XI.7" -> "I.def.4"; "XI.8" -> "I.def.4"; "XI.9" -> "I.def.4"; "XI.10" -> "I.def.4"; "XI.29" -> "I.def.4"; "XI.30" -> "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"; "XI.def.3" -> "I.def.7"; "XI.def.12" -> "I.def.7"; "XI.def.13" -> "I.def.7"; "XI.1" -> "I.def.7"; "XI.2" -> "I.def.7"; "XI.3" -> "I.def.7"; "XI.5" -> "I.def.7"; "XI.7" -> "I.def.7"; "XI.8" -> "I.def.7"; "XI.9" -> "I.def.7"; "XI.10" -> "I.def.7"; "XI.16" -> "I.def.7"; "XI.24" -> "I.def.7"; "XI.25" -> "I.def.7"; "I.def.9" -> "I.def.8"; "I.def.10" -> "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.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.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.34" -> "I.def.8"; "XI.10" -> "I.def.8"; "I.9" -> "I.def.9"; "I.23" -> "I.def.9"; "I.42" -> "I.def.9"; "I.44" -> "I.def.9"; "I.45" -> "I.def.9"; "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.28" -> "I.def.10"; "I.29" -> "I.def.10"; "VI.def.4" -> "I.def.10"; "XI.def.3" -> "I.def.10"; "XI.4" -> "I.def.10"; "XI.5" -> "I.def.10"; "XI.8" -> "I.def.10"; "I.def.14" -> "I.def.13"; "I.def.15" -> "I.def.14"; "I.4" -> "I.def.14"; "V.9" -> "I.def.14"; "V.11" -> "I.def.14"; "VI.def.4" -> "I.def.14"; "I.def.16" -> "I.def.15"; "I.def.17" -> "I.def.15"; "I.post.3" -> "I.def.15"; "I.1" -> "I.def.15"; "I.3" -> "I.def.15"; "XI.1" -> "I.def.15"; "I.def.17" -> "I.def.16"; "I.post.3" -> "I.def.16"; "XI.1" -> "I.def.16"; "I.34" -> "I.def.17"; "I.43" -> "I.def.17"; "XI.1" -> "I.def.17"; "I.def.20" -> "I.def.19"; "I.def.22" -> "I.def.19"; "I.4" -> "I.def.19"; "I.8" -> "I.def.19"; "I.16" -> "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.37" -> "I.def.19"; "I.38" -> "I.def.19"; "I.41" -> "I.def.19"; "I.42" -> "I.def.19"; "I.44" -> "I.def.19"; "I.45" -> "I.def.19"; "VI.1" -> "I.def.19"; "XI.2" -> "I.def.19"; "I.def.22" -> "I.def.20"; "I.1" -> "I.def.20"; "I.5" -> "I.def.20"; "I.35" -> "I.def.22"; "I.27" -> "I.def.23"; "I.28" -> "I.def.23"; "I.29" -> "I.def.23"; "I.30" -> "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.41" -> "I.def.23"; "XI.def.13" -> "I.def.23"; "XI.6" -> "I.def.23"; "XI.7" -> "I.def.23"; "XI.8" -> "I.def.23"; "XI.9" -> "I.def.23"; "XI.10" -> "I.def.23"; "XI.16" -> "I.def.23"; "XI.25" -> "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.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"; "I.44" -> "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.30" -> "I.cn.1"; "I.35" -> "I.cn.1"; "I.36" -> "I.cn.1"; "I.44" -> "I.cn.1"; "I.45" -> "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.45" -> "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"; "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"; "XI.4" -> "I.4"; "XI.6" -> "I.4"; "XI.24" -> "I.4"; "XI.29" -> "I.4"; "I.7" -> "I.5"; "I.19" -> "I.5"; "I.20" -> "I.5"; "I.8" -> "I.7"; "I.9" -> "I.8"; "I.11" -> "I.8"; "I.23" -> "I.8"; "XI.4" -> "I.8"; "XI.6" -> "I.8"; "XI.10" -> "I.8"; "XI.29" -> "I.8"; "I.10" -> "I.9"; "I.16" -> "I.10"; "I.13" -> "I.11"; "I.14" -> "I.13"; "I.15" -> "I.13"; "I.28" -> "I.13"; "I.29" -> "I.13"; "I.45" -> "I.14"; "I.16" -> "I.15"; "I.28" -> "I.15"; "I.29" -> "I.15"; "I.44" -> "I.15"; "XI.4" -> "I.15"; "I.18" -> "I.16"; "I.26" -> "I.16"; "I.27" -> "I.16"; "I.19" -> "I.18"; "I.20" -> "I.19"; "I.22" -> "I.20"; "I.23" -> "I.22"; "I.31" -> "I.23"; "I.42" -> "I.23"; "XI.31" -> "I.23"; "I.34" -> "I.26"; "XI.4" -> "I.26"; "I.28" -> "I.27"; "I.31" -> "I.27"; "I.33" -> "I.27"; "XI.6" -> "I.28"; "I.30" -> "I.29"; "I.33" -> "I.29"; "I.34" -> "I.29"; "I.35" -> "I.29"; "I.44" -> "I.29"; "I.45" -> "I.29"; "XI.8" -> "I.29"; "I.45" -> "I.30"; "I.37" -> "I.31"; "I.38" -> "I.31"; "I.42" -> "I.31"; "I.44" -> "I.31"; "I.36" -> "I.33"; "I.45" -> "I.33"; "XI.10" -> "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.45" -> "I.34"; "XI.24" -> "I.34"; "XI.29" -> "I.34"; "XII.9" -> "I.34"; "I.36" -> "I.35"; "I.37" -> "I.35"; "XI.31" -> "I.35"; "I.38" -> "I.36"; "XI.29" -> "I.36"; "I.41" -> "I.37"; "I.42" -> "I.38"; "VI.1" -> "I.38"; "I.42" -> "I.41"; "VI.1" -> "I.41"; "I.44" -> "I.42"; "I.45" -> "I.42"; "I.44" -> "I.43"; "I.45" -> "I.44"; "XI.32" -> "I.45"; "V.def.2" -> "V.def.1"; "VII.def.3" -> "V.def.1"; "VII.def.5" -> "V.def.1"; "V.def.5" -> "V.def.2"; "V.def.7" -> "V.def.2"; "V.1" -> "V.def.2"; "V.15" -> "V.def.2"; "V.11" -> "V.def.3"; "V.8" -> "V.def.4"; "V.def.6" -> "V.def.5"; "V.7" -> "V.def.5"; "V.9" -> "V.def.5"; "V.11" -> "V.def.5"; "V.12" -> "V.def.5"; "V.15" -> "V.def.5"; "VI.1" -> "V.def.5"; "XI.25" -> "V.def.5"; "V.7" -> "V.def.6"; "V.12" -> "V.def.6"; "V.8" -> "V.def.7"; "V.def.13" -> "V.def.11"; "V.12" -> "V.def.11"; "V.7" -> "V.def.13"; "V.8" -> "V.1"; "V.12" -> "V.1"; "V.15" -> "V.7"; "XI.31" -> "V.7"; "XI.34" -> "V.7"; "V.9" -> "V.8"; "XI.31" -> "V.9"; "XI.34" -> "V.9"; "VI.1" -> "V.11"; "XI.31" -> "V.11"; "XII.9" -> "V.11"; "V.15" -> "V.12"; "VI.1" -> "V.15"; "XI.34" -> "VI.def.2"; "XII.9" -> "VI.def.2"; "VI.1" -> "VI.def.4"; "XI.29" -> "VI.def.4"; "XI.30" -> "VI.def.4"; "XI.31" -> "VI.def.4"; "XI.32" -> "VI.def.4"; "XI.34" -> "VI.def.4"; "XII.9" -> "VI.def.4"; "XI.34" -> "VI.1"; "VII.def.2" -> "VII.def.1"; "VII.def.3" -> "VII.def.2"; "VII.def.5" -> "VII.def.2"; "XI.1" -> "VII.def.3"; "XI.25" -> "VII.def.5"; "XI.def.2" -> "XI.def.1"; "XI.def.12" -> "XI.def.1"; "XI.def.13" -> "XI.def.1"; "XI.24" -> "XI.def.1"; "XI.25" -> "XI.def.1"; "XI.29" -> "XI.def.1"; "XI.30" -> "XI.def.1"; "XI.31" -> "XI.def.1"; "XI.32" -> "XI.def.1"; "XI.34" -> "XI.def.1"; "XI.def.10" -> "XI.def.2"; "XI.4" -> "XI.def.3"; "XI.5" -> "XI.def.3"; "XI.6" -> "XI.def.3"; "XI.8" -> "XI.def.3"; "XI.16" -> "XI.def.8"; "XI.24" -> "XI.def.8"; "XI.31" -> "XI.def.10"; "XII.9" -> "XI.def.12"; "XI.29" -> "XI.def.13"; "XI.2" -> "XI.1"; "XI.16" -> "XI.1"; "XI.6" -> "XI.2"; "XI.8" -> "XI.2"; "XI.5" -> "XI.3"; "XI.7" -> "XI.3"; "XI.5" -> "XI.4"; "XI.8" -> "XI.4"; "XI.9" -> "XI.4"; "XI.6" -> "XI.5"; "XI.9" -> "XI.6"; "XI.8" -> "XI.7"; "XI.9" -> "XI.8"; "XI.10" -> "XI.9"; "XI.24" -> "XI.10"; "XI.24" -> "XI.16"; "XI.25" -> "XI.24"; "XI.31" -> "XI.24"; "XI.31" -> "XI.25"; "XI.32" -> "XI.25"; "XI.34" -> "XI.25"; "XI.30" -> "XI.29"; "XI.31" -> "XI.29"; "XI.34" -> "XI.29"; "XI.31" -> "XI.30"; "XI.34" -> "XI.30"; "XI.32" -> "XI.31"; "XI.34" -> "XI.31"; "XI.34" -> "XI.32"; "XII.9" -> "XI.34"; }

See Commentary on Proposition XII.9.