PROPOSITION 3.

Any pyramid which has a triangular base is divided into two pyramids equal and similar to one another, similar to the whole and having triangular bases, and into two equal prisms; and the two prisms are greater than the half of the whole pyramid.

Let there be a pyramid of which the triangle ABC is the base and the point D the vertex; I say that the pyramid ABCD is divided into two pyramids equal to one another, having triangular bases and similar to the whole pyramid, and into two equal prisms; and the two prisms are greater than the half of the whole pyramid.

D L H K C G F A E B

For let AB, BC, CA, AD, DB, DC be bisected at the points E, F, G, H, K, L, and let HE, EG, GH, HK, KL, LH, KF, FG be joined.

Since AE is equal to EB, and AH to DH,
[VI. 2]therefore EH is parallel to DB.
For the same reason HK is also parallel to AB.
Therefore HEBK is a parallelogram; therefore HK is equal to EB. [I. 34]
But EB is equal to EA;
therefore AE is also equal to HK.
But AH is also equal to HD;
therefore the two sides EA, AH are equal to the two sides KH, HD respectively, and the angle EAH is equal to the angle KHD; therefore the base EH is equal to the base KD. [I. 4]
Therefore the triangle AEH is equal and similar to the triangle HKD.
For the same reason the triangle AHG is also equal and similar to the triangle HLD.
Now, since two straight lines EH, HG meeting one another are parallel to two straight lines KD, DL meeting one another, and are not in the same plane, they will contain equal angles. [XI. 10]
Therefore the angle EHG is equal to the angle KDL.
And, since the two straight lines EH, HG are equal to the two KD, DL respectively, and the angle EHG is equal to the angle KDL,
therefore the base EG is equal to the base KL; [I. 4] therefore the triangle EHG is equal and similar to the triangle KDL.
For the same reason the triangle AEG is also equal and similar to the triangle HKL.
Therefore the pyramid of which the triangle AEG is the base and the point H the vertex is equal and similar to the pyramid of which the triangle HKL is the base and the point D the vertex. [XI. Def. 10]
And, since HK has been drawn parallel to AB, one of the sides of the triangle ADB,
the triangle ADB is equiangular to the triangle DHK, [I. 29] and they have their sides proportional; therefore the triangle ADB is similar to the triangle DHK. [VI. Def. 1]
For the same reason the triangle DBC is also similar to the triangle DKL, and the triangle ADC to the triangle DLH.
Now, since the two straight lines BA, AC meeting one another are parallel to the two straight lines KH, HL meeting one another, not in the same plane, they will contain equal angles. [XI. 10]
Therefore the angle BAC is equal to the angle KHL.
And, as BA is to AC, so is KH to HL;
therefore the triangle ABC is similar to the triangle HKL.
Therefore also the pyramid of which the triangle ABC is the base and the point D the vertex is similar to the pyramid of which the triangle HKL is the base and the point D the vertex.
But the pyramid of which the triangle HKL is the base and the point D the vertex was proved similar to the pyramid of which the triangle AEG is the base and the point H the vertex.
Therefore each of the pyramids AEGH, HKLD is similar to the whole pyramid ABCD.
Next, since BF is equal to FC,
the parallelogram EBFG is double of the triangle GFC.
And since, if there be two prisms of equal height, and one have a parallelogram as base, and the other a triangle, and if the parallelogram be double of the triangle, the prisms are equal, [XI. 39]
therefore the prism contained by the two triangles BKF, EHG, and the three parallelograms EBFG, EBKH, HKFG is equal to the prism contained by the two triangles GFC, HKL and the three parallelograms KFCL, LCGH, HKFG.
And it is manifest that each of the prisms, namely that in which the parallelogram EBFG is the base and the straight line HK is its opposite, and that in which the triangle GFC is the base and the triangle HKL its opposite, is greater than each of the pyramids of which the triangles AEG, HKL are the bases and the points H, D the vertices,
inasmuch as, if we join the straight lines EF, EK, the prism in which the parallelogram EBFG is the base and the straight line HK its opposite is greater than the pyramid of which the triangle EBF is the base and the point K the vertex.
But the pyramid of which the triangle EBF is the base and the point K the vertex is equal to the pyramid of which the triangle AEG is the base and the point H the vertex; for they are contained by equal and similar planes.
Hence also the prism in which the parallelogram EBFG is the base and the straight line HK its opposite is greater than the pyramid of which the triangle AEG is the base and the point H the vertex.
But the prism in which the parallelogram EBFG is the base and the straight line HK its opposite is equal to the prism in which the triangle GFC is the base and the triangle HKL its opposite, and the pyramid of which the triangle AEG is the base and the point H the vertex is equal to the pyramid of which the triangle HKL is the base and the point D the vertex.
Therefore the said two prisms are greater than the said two pyramids of which the triangles AEG, HKL are the bases and the points H, D the vertices.
Therefore the whole pyramid, of which the triangle ABC is the base and the point D the vertex, has been divided into two pyramids equal to one another and into two equal prisms, and the two prisms are greater than the half of the whole pyramid.

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.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.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.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"]; "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.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.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"]; "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.28" [label="XI.28", fillcolor="#222244", tooltip="XI.28: If a parallelepipedal solid be cut by a plane through the diagonals of the opposite planes, the solid will be bisected by the plane.", URL="../../../book_xi/propositions/28/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.39" [label="XI.39", fillcolor="#222244", tooltip="XI.39: If there be two prisms of equal height, and one have a parallelogram as base and the other a triangle, and if the parallelogram be double of the triangle, the prisms will be equal.", URL="../../../book_xi/propositions/39/index.html", target="_top"]; "XII.3" [label="XII.3", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="XII.3: Any pyramid which has a triangular base is divided into two pyramids equal and similar to one another, similar to the whole and having triangular bases, and into two equal prisms…", 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.23" -> "I.def.1"; "I.31" -> "I.def.1"; "VI.2" -> "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"; "VI.2" -> "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.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"; "VI.2" -> "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"; "XI.28" -> "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.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"; "VI.def.1" -> "I.def.8"; "XI.10" -> "I.def.8"; "I.9" -> "I.def.9"; "I.23" -> "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.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"; "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.39" -> "I.def.19"; "I.41" -> "I.def.19"; "VI.1" -> "I.def.19"; "VI.2" -> "I.def.19"; "XI.2" -> "I.def.19"; "XI.39" -> "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.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"; "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.15" -> "I.post.4"; "I.29" -> "I.post.5"; "I.1" -> "I.cn.1"; "I.2" -> "I.cn.1"; "I.3" -> "I.cn.1"; "I.13" -> "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.2" -> "I.cn.3"; "I.15" -> "I.cn.3"; "I.35" -> "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"; "XII.3" -> "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.15" -> "I.13"; "I.28" -> "I.13"; "I.29" -> "I.13"; "I.16" -> "I.15"; "I.28" -> "I.15"; "I.29" -> "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"; "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.33" -> "I.29"; "I.34" -> "I.29"; "I.35" -> "I.29"; "XI.8" -> "I.29"; "XII.3" -> "I.29"; "I.37" -> "I.31"; "I.38" -> "I.31"; "I.39" -> "I.31"; "I.36" -> "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"; "XI.24" -> "I.34"; "XI.28" -> "I.34"; "XI.29" -> "I.34"; "XI.39" -> "I.34"; "XII.3" -> "I.34"; "I.36" -> "I.35"; "I.37" -> "I.35"; "XI.31" -> "I.35"; "I.38" -> "I.36"; "XI.29" -> "I.36"; "I.39" -> "I.37"; "I.41" -> "I.37"; "VI.1" -> "I.38"; "VI.2" -> "I.38"; "VI.2" -> "I.39"; "VI.1" -> "I.41"; "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"; "VI.def.1" -> "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"; "VI.2" -> "V.7"; "XI.31" -> "V.7"; "V.9" -> "V.8"; "VI.2" -> "V.9"; "XI.31" -> "V.9"; "VI.1" -> "V.11"; "VI.2" -> "V.11"; "XI.31" -> "V.11"; "V.15" -> "V.12"; "VI.1" -> "V.15"; "XII.3" -> "VI.def.1"; "VI.1" -> "VI.def.4"; "XI.29" -> "VI.def.4"; "XI.30" -> "VI.def.4"; "XI.31" -> "VI.def.4"; "XI.39" -> "VI.def.4"; "VI.2" -> "VI.1"; "XII.3" -> "VI.2"; "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.28" -> "XI.def.1"; "XI.29" -> "XI.def.1"; "XI.30" -> "XI.def.1"; "XI.31" -> "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.28" -> "XI.def.10"; "XI.31" -> "XI.def.10"; "XII.3" -> "XI.def.10"; "XII.3" -> "XI.def.12"; "XI.28" -> "XI.def.13"; "XI.29" -> "XI.def.13"; "XI.39" -> "XI.def.13"; "XII.3" -> "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"; "XII.3" -> "XI.10"; "XI.24" -> "XI.16"; "XI.25" -> "XI.24"; "XI.31" -> "XI.24"; "XI.31" -> "XI.25"; "XI.39" -> "XI.28"; "XI.30" -> "XI.29"; "XI.31" -> "XI.29"; "XI.31" -> "XI.30"; "XI.39" -> "XI.31"; "XII.3" -> "XI.39"; }

Required for

XII.4, XII.5, XII.6, XII.7, XII.8, XII.10, XII.11, XII.12, XII.13, XII.14, XII.15, XII.17, XII.18

See Commentary on Proposition XII.3.