PROPOSITION 13.

If a cylinder be cut by a plane which is parallel to its opposite planes, then, as the cylinder is to the cylinder, so will the axis be to the axis.

For let the cylinder AD be cut by the plane GH which is parallel to the opposite planes AB, CD, and let the plane GH meet the axis at the point K; I say that, as the cylinder BG is to the cylinder GD, so is the axis EK to the axis KF.

P L Q R N S A E B G K H C F D T O U V M W

For let the axis EF be produced in both directions to the points L, M, and let there be set out any number whatever of axes EN, NL equal to the axis EK, and any number whatever FO, OM equal to FK; and let the cylinder PW on the axis LM be conceived of which the circles PQ, VW are the bases.

Let planes be carried through the points N, O parallel to AB, CD and to the bases of the cylinder PW, and let them produce the circles RS, TU about the centres N, O.

Then, since the axes LN, NE, EK are equal to one another,
therefore the cylinders QR, RB, BG are to one another as their bases. [XII. 11]
But the bases are equal;
therefore the cylinders QR, RB, BG are also equal to one another.
Since then the axes LN, NE, EK are equal to one another, and the cylinders QR, RB, BG are also equal to one another, and the multitude of the former is equal to the multitude of the latter,
therefore, whatever multiple the axis KL is of the axis EK, the same multiple also will the cylinder QG be of the cylinder GB.
For the same reason, whatever multiple the axis MK is of the axis KF, the same multiple also is the cylinder WG of the cylinder GD.
And, if the axis KL is equal to the axis KM, the cylinder QG will also be equal to the cylinder GW, if the axis is greater than the axis, the cylinder will also be greater than the cylinder, and if less, less.
Thus, there being four magnitudes, the axes EK, KF and the cylinders BG, GD, there have been taken equimultiples of the axis EK and of the cylinder BG, namely the axis LK and the cylinder QG, and equimultiples of the axis KF and of the cylinder GD, namely the axis KM and the cylinder GW; and it has been proved that, if the axis KL is in excess of the axis KM, the cylinder QG is also in excess of the cylinder GW, if equal, equal, and if less, less.
Therefore, as the axis EK is to the axis KF, so is the cylinder BG to the cylinder GD. [V. Def. 5]
Q. E. D.

Dependency Graph

digraph { bgcolor="transparent"; node [shape=box, style="rounded,filled", fontname="Helvetica", fontsize=11, color="#d6d3cd", fontcolor="#ffffff"]; edge [color="#8c857b", fontcolor="#8c857b", arrowsize=0.75]; rankdir="TB"; "I.def.1" [label="I.def.1", fillcolor="#224422", tooltip="I.def.1: A point is that which has no part.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-1", target="_top"]; "I.def.2" [label="I.def.2", fillcolor="#224422", tooltip="I.def.2: A line is breadthless length.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-2", target="_top"]; "I.def.3" [label="I.def.3", fillcolor="#224422", tooltip="I.def.3: The extremities of a line are points.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-3", target="_top"]; "I.def.4" [label="I.def.4", fillcolor="#224422", tooltip="I.def.4: A straight line is a line which lies evenly with the points on itself.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-4", target="_top"]; "I.def.5" [label="I.def.5", fillcolor="#224422", tooltip="I.def.5: A surface is that which has length and breadth only.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-5", target="_top"]; "I.def.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.11" [label="I.def.11", fillcolor="#224422", tooltip="I.def.11: An obtuse angle is an angle greater than a right angle.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-11", target="_top"]; "I.def.12" [label="I.def.12", fillcolor="#224422", tooltip="I.def.12: An acute angle is an angle less than a right angle.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-12", 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.18" [label="I.def.18", fillcolor="#224422", tooltip="I.def.18: A semicircle is the figure contained by the diameter and the circumference cut off by it. 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.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.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.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.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"]; "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"]; "III.def.1" [label="III.def.1", fillcolor="#224422", tooltip="III.def.1: Equal circles are those the diameters of which are equal, or the radii of which are equal.", URL="../../../../volume_II/book_iii/definitions/index.html#iii-def-1", target="_top"]; "III.def.2" [label="III.def.2", fillcolor="#224422", tooltip="III.def.2: A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.", URL="../../../../volume_II/book_iii/definitions/index.html#iii-def-2", target="_top"]; "III.def.6" [label="III.def.6", fillcolor="#224422", tooltip="III.def.6: A segment of a circle is the figure contained by a straight line and a circumference of a circle.", URL="../../../../volume_II/book_iii/definitions/index.html#iii-def-6", target="_top"]; "III.def.7" [label="III.def.7", fillcolor="#224422", tooltip="III.def.7: An angle of a segment is that contained by a straight line and a circumference of a circle.", URL="../../../../volume_II/book_iii/definitions/index.html#iii-def-7", 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.9" [label="III.def.9", fillcolor="#224422", tooltip="III.def.9: And, when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference.", URL="../../../../volume_II/book_iii/definitions/index.html#iii-def-9", 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.def.11" [label="III.def.11", fillcolor="#224422", tooltip="III.def.11: Similar segments of circles are those which admit equal angles, or in which the angles are equal to one another.", URL="../../../../volume_II/book_iii/definitions/index.html#iii-def-11", target="_top"]; "III.1" [label="III.1", fillcolor="#222244", tooltip="III.1: To find the centre of a given circle.", URL="../../../../volume_II/book_iii/propositions/01/index.html", target="_top"]; "III.5" [label="III.5", fillcolor="#222244", tooltip="III.5: If two circles cut one another, they will not have the same centre.", URL="../../../../volume_II/book_iii/propositions/05/index.html", target="_top"]; "III.10" [label="III.10", fillcolor="#222244", tooltip="III.10: A circle does not cut a circle at more points than two.", URL="../../../../volume_II/book_iii/propositions/10/index.html", target="_top"]; "III.16" [label="III.16", fillcolor="#222244", tooltip="III.16: The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the…", URL="../../../../volume_II/book_iii/propositions/16/index.html", target="_top"]; "III.18" [label="III.18", fillcolor="#222244", tooltip="III.18: If a straight line touch a circle, and a straight line be joined from the centre to the point of contact, the straight line so joined will be perpendicular to the tangent.", URL="../../../../volume_II/book_iii/propositions/18/index.html", 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.24" [label="III.24", fillcolor="#222244", tooltip="III.24: Similar segments of circles on equal straight lines are equal to one another.", URL="../../../../volume_II/book_iii/propositions/24/index.html", target="_top"]; "III.26" [label="III.26", fillcolor="#222244", tooltip="III.26: In equal circles equal angles stand on equal circumferences, whether they stand at the centres or at the circumferences.", URL="../../../../volume_II/book_iii/propositions/26/index.html", target="_top"]; "III.27" [label="III.27", fillcolor="#222244", tooltip="III.27: In equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centres or at the circumferences.", URL="../../../../volume_II/book_iii/propositions/27/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.3" [label="IV.def.3", fillcolor="#224422", tooltip="IV.def.3: A rectilineal figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle.", URL="../../../../volume_II/book_iv/definitions/index.html#iv-def-3", target="_top"]; "IV.def.4" [label="IV.def.4", fillcolor="#224422", tooltip="IV.def.4: A rectilineal figure is said to be circumscribed about a circle, when each side of the circumscribed figure touches the circumference of the circle.", URL="../../../../volume_II/book_iv/definitions/index.html#iv-def-4", target="_top"]; "IV.6" [label="IV.6", fillcolor="#222244", tooltip="IV.6: In a given circle to inscribe a square.", URL="../../../../volume_II/book_iv/propositions/06/index.html", target="_top"]; "IV.7" [label="IV.7", fillcolor="#222244", tooltip="IV.7: About a given circle to circumscribe a square.", URL="../../../../volume_II/book_iv/propositions/07/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.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.18" [label="V.18", fillcolor="#222244", tooltip="V.18: If magnitudes be proportional separando, they will also be proportional componendo.", URL="../../../../volume_II/book_v/propositions/18/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.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.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.6" [label="VII.def.6", fillcolor="#224422", tooltip="VII.def.6: An even number is that which is divisible into two equal parts.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-6", 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"]; "X.1" [label="X.1", fillcolor="#222244", tooltip="X.1: Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and…", URL="../../../book_x/propositions/001/index.html", 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.def.18" [label="XI.def.18", fillcolor="#224422", tooltip="XI.def.18: When, one side of those about the right angle in a right-angled triangle remaining fixed, the triangle is carried round and restored again to the same position from which it began…", URL="../../../book_xi/definitions/index.html#xi-def-18", target="_top"]; "XI.def.19" [label="XI.def.19", fillcolor="#224422", tooltip="XI.def.19: The axis of the cone is the straight line which remains fixed and about which the triangle is turned.", URL="../../../book_xi/definitions/index.html#xi-def-19", target="_top"]; "XI.def.21" [label="XI.def.21", fillcolor="#224422", tooltip="XI.def.21: When, one side of those about the right angle in a rectangular parallelogram remaining fixed, the parallelogram is carried round and restored again to the same position from which…", URL="../../../book_xi/definitions/index.html#xi-def-21", target="_top"]; "XI.def.23" [label="XI.def.23", fillcolor="#224422", tooltip="XI.def.23: And the bases are the circles described by the two sides opposite to one another which are carried round.", URL="../../../book_xi/definitions/index.html#xi-def-23", 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.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.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.1" [label="XII.1", fillcolor="#222244", tooltip="XII.1: Similar polygons inscribed in circles are to one another as the squares on the diameters.", URL="../01/index.html", target="_top"]; "XII.2" [label="XII.2", fillcolor="#222244", tooltip="XII.2: Circles are to one another as the squares on the diameters.", URL="../02/index.html", target="_top"]; "XII.3" [label="XII.3", fillcolor="#222244", 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="../03/index.html", target="_top"]; "XII.4" [label="XII.4", fillcolor="#222244", tooltip="XII.4: If there be two pyramids of the same height which have triangular bases, and each of them be divided into two pyramids equal to one another and similar to the whole, and into two…", URL="../04/index.html", target="_top"]; "XII.5" [label="XII.5", fillcolor="#222244", tooltip="XII.5: Pyramids which are of the same height and have triangular bases are to one another as the bases.", URL="../05/index.html", target="_top"]; "XII.6" [label="XII.6", fillcolor="#222244", tooltip="XII.6: Pyramids which are of the same height and have polygonal bases are to one another as the bases.", URL="../06/index.html", target="_top"]; "XII.7" [label="XII.7", fillcolor="#222244", tooltip="XII.7: Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases.", URL="../07/index.html", target="_top"]; "XII.10" [label="XII.10", fillcolor="#222244", tooltip="XII.10: Any cone is a third part of the cylinder which has the same base with it and equal height.", URL="../10/index.html", target="_top"]; "XII.11" [label="XII.11", fillcolor="#222244", tooltip="XII.11: Cones and cylinders which are of the same height are to one another as their bases.", URL="../11/index.html", target="_top"]; "XII.13" [label="XII.13", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="XII.13: If a cylinder be cut by a plane which is parallel to its opposite planes, then, as the cylinder is to the cylinder, so will the axis be to the axis.", 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"; "III.def.8" -> "I.def.1"; "III.10" -> "I.def.1"; "III.18" -> "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"; "V.17" -> "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"; "III.def.8" -> "I.def.3"; "III.16" -> "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"; "II.def.1" -> "I.def.4"; "III.def.2" -> "I.def.4"; "III.def.6" -> "I.def.4"; "III.def.7" -> "I.def.4"; "III.def.8" -> "I.def.4"; "III.def.9" -> "I.def.4"; "III.def.10" -> "I.def.4"; "III.16" -> "I.def.4"; "III.18" -> "I.def.4"; "III.24" -> "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"; "XI.def.3" -> "I.def.4"; "XI.def.19" -> "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"; "XII.13" -> "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.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.9" -> "I.def.8"; "III.def.10" -> "I.def.8"; "III.def.11" -> "I.def.8"; "III.16" -> "I.def.8"; "III.20" -> "I.def.8"; "III.21" -> "I.def.8"; "III.22" -> "I.def.8"; "III.26" -> "I.def.8"; "III.27" -> "I.def.8"; "III.31" -> "I.def.8"; "IV.def.3" -> "I.def.8"; "VI.def.1" -> "I.def.8"; "VI.4" -> "I.def.8"; "VI.6" -> "I.def.8"; "VI.15" -> "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"; "III.16" -> "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"; "II.def.1" -> "I.def.10"; "III.1" -> "I.def.10"; "III.16" -> "I.def.10"; "III.18" -> "I.def.10"; "III.22" -> "I.def.10"; "III.31" -> "I.def.10"; "VI.def.4" -> "I.def.10"; "XI.def.3" -> "I.def.10"; "XI.def.18" -> "I.def.10"; "XI.def.21" -> "I.def.10"; "XI.4" -> "I.def.10"; "XI.5" -> "I.def.10"; "XI.8" -> "I.def.10"; "I.def.21" -> "I.def.11"; "I.def.21" -> "I.def.12"; "III.16" -> "I.def.12"; "III.18" -> "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"; "III.def.6" -> "I.def.14"; "III.def.10" -> "I.def.14"; "IV.def.3" -> "I.def.14"; "IV.def.4" -> "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"; "XI.def.18" -> "I.def.14"; "XI.def.21" -> "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.2" -> "I.def.15"; "III.def.6" -> "I.def.15"; "III.def.7" -> "I.def.15"; "III.def.10" -> "I.def.15"; "III.1" -> "I.def.15"; "III.5" -> "I.def.15"; "III.10" -> "I.def.15"; "III.16" -> "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.3" -> "I.def.15"; "IV.def.4" -> "I.def.15"; "IV.6" -> "I.def.15"; "IV.7" -> "I.def.15"; "XI.def.23" -> "I.def.15"; "XI.1" -> "I.def.15"; "XII.2" -> "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"; "III.1" -> "I.def.16"; "III.5" -> "I.def.16"; "III.18" -> "I.def.16"; "III.26" -> "I.def.16"; "III.27" -> "I.def.16"; "XI.1" -> "I.def.16"; "I.def.18" -> "I.def.17"; "I.34" -> "I.def.17"; "I.43" -> "I.def.17"; "III.def.1" -> "I.def.17"; "III.16" -> "I.def.17"; "III.31" -> "I.def.17"; "XI.1" -> "I.def.17"; "XII.1" -> "I.def.17"; "XII.2" -> "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.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"; "I.42" -> "I.def.19"; "I.44" -> "I.def.19"; "I.45" -> "I.def.19"; "III.22" -> "I.def.19"; "IV.def.3" -> "I.def.19"; "IV.def.4" -> "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"; "XI.def.18" -> "I.def.19"; "XI.def.19" -> "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"; "XI.def.18" -> "I.def.21"; "I.35" -> "I.def.22"; "IV.6" -> "I.def.22"; "IV.7" -> "I.def.22"; "XII.1" -> "I.def.22"; "XII.2" -> "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.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"; "XII.13" -> "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"; "I.44" -> "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.30" -> "I.cn.1"; "I.35" -> "I.cn.1"; "I.36" -> "I.cn.1"; "I.39" -> "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"; "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"; "III.26" -> "I.4"; "IV.6" -> "I.4"; "VI.6" -> "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"; "III.16" -> "I.5"; "III.20" -> "I.5"; "III.31" -> "I.5"; "I.8" -> "I.7"; "I.9" -> "I.8"; "I.11" -> "I.8"; "I.23" -> "I.8"; "III.1" -> "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.17" -> "I.13"; "I.28" -> "I.13"; "I.29" -> "I.13"; "I.32" -> "I.13"; "I.45" -> "I.14"; "VI.15" -> "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"; "III.16" -> "I.17"; "III.18" -> "I.17"; "III.31" -> "I.17"; "VI.4" -> "I.17"; "I.19" -> "I.18"; "I.20" -> "I.19"; "III.16" -> "I.19"; "III.18" -> "I.19"; "I.22" -> "I.20"; "I.23" -> "I.22"; "I.31" -> "I.23"; "I.42" -> "I.23"; "III.27" -> "I.23"; "VI.6" -> "I.23"; "VI.18" -> "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"; "IV.7" -> "I.28"; "VI.4" -> "I.28"; "XI.6" -> "I.28"; "I.30" -> "I.29"; "I.32" -> "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"; "XII.3" -> "I.29"; "I.45" -> "I.30"; "IV.7" -> "I.30"; "I.32" -> "I.31"; "I.37" -> "I.31"; "I.38" -> "I.31"; "I.39" -> "I.31"; "I.42" -> "I.31"; "I.44" -> "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"; "XII.1" -> "I.32"; "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"; "IV.7" -> "I.34"; "VI.4" -> "I.34"; "XI.24" -> "I.34"; "XI.28" -> "I.34"; "XI.29" -> "I.34"; "XI.39" -> "I.34"; "XII.3" -> "I.34"; "XII.7" -> "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"; "I.42" -> "I.38"; "VI.1" -> "I.38"; "VI.2" -> "I.38"; "VI.2" -> "I.39"; "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"; "XI.def.21" -> "II.def.1"; "III.26" -> "III.def.1"; "III.27" -> "III.def.1"; "III.16" -> "III.def.2"; "III.18" -> "III.def.2"; "XII.2" -> "III.def.6"; "III.16" -> "III.def.7"; "III.31" -> "III.def.8"; "III.26" -> "III.def.9"; "III.27" -> "III.def.9"; "III.20" -> "III.def.10"; "III.24" -> "III.def.11"; "III.26" -> "III.def.11"; "III.10" -> "III.1"; "III.10" -> "III.5"; "III.24" -> "III.10"; "IV.7" -> "III.16"; "IV.7" -> "III.18"; "III.21" -> "III.20"; "III.27" -> "III.20"; "III.22" -> "III.21"; "III.31" -> "III.22"; "III.26" -> "III.24"; "III.27" -> "III.26"; "XII.1" -> "III.27"; "IV.6" -> "III.31"; "XII.1" -> "III.31"; "IV.6" -> "IV.def.3"; "XII.1" -> "IV.def.3"; "IV.7" -> "IV.def.4"; "XII.2" -> "IV.def.4"; "XII.10" -> "IV.6"; "XII.10" -> "IV.7"; "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.2" -> "V.def.2"; "V.3" -> "V.def.2"; "V.4" -> "V.def.2"; "V.15" -> "V.def.2"; "V.def.17" -> "V.def.3"; "V.10" -> "V.def.3"; "V.11" -> "V.def.3"; "VI.20" -> "V.def.3"; "V.8" -> "V.def.4"; "V.10" -> "V.def.4"; "X.1" -> "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"; "XI.25" -> "V.def.5"; "XII.13" -> "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"; "V.18" -> "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"; "V.8" -> "V.def.7"; "V.10" -> "V.def.7"; "V.13" -> "V.def.7"; "VI.19" -> "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.12" -> "V.def.11"; "VI.19" -> "V.def.11"; "V.16" -> "V.def.12"; "V.7" -> "V.def.13"; "V.17" -> "V.def.14"; "V.18" -> "V.def.14"; "V.17" -> "V.def.15"; "V.18" -> "V.def.15"; "V.3" -> "V.def.17"; "V.20" -> "V.def.17"; "V.22" -> "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.15" -> "V.7"; "XI.31" -> "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.15" -> "V.9"; "VI.22" -> "V.9"; "XI.31" -> "V.9"; "V.14" -> "V.10"; "V.20" -> "V.10"; "V.16" -> "V.11"; "V.18" -> "V.11"; "VI.1" -> "V.11"; "VI.2" -> "V.11"; "VI.6" -> "V.11"; "VI.15" -> "V.11"; "VI.19" -> "V.11"; "VI.22" -> "V.11"; "XI.31" -> "V.11"; "XII.2" -> "V.11"; "XII.5" -> "V.11"; "XII.11" -> "V.11"; "V.15" -> "V.12"; "VI.20" -> "V.12"; "XII.4" -> "V.12"; "V.14" -> "V.13"; "V.20" -> "V.13"; "V.16" -> "V.14"; "V.18" -> "V.14"; "V.16" -> "V.15"; "VI.1" -> "V.15"; "VI.4" -> "V.16"; "VI.19" -> "V.16"; "XII.2" -> "V.16"; "XII.4" -> "V.16"; "XII.5" -> "V.16"; "XII.11" -> "V.16"; "V.18" -> "V.17"; "XII.6" -> "V.18"; "V.22" -> "V.20"; "VI.4" -> "V.22"; "VI.20" -> "V.22"; "VI.22" -> "V.22"; "XII.6" -> "V.22"; "VI.18" -> "VI.def.1"; "VI.19" -> "VI.def.1"; "VI.20" -> "VI.def.1"; "VI.22" -> "VI.def.1"; "XII.1" -> "VI.def.1"; "XII.3" -> "VI.def.1"; "VI.15" -> "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.39" -> "VI.def.4"; "XII.4" -> "VI.def.4"; "XII.5" -> "VI.def.4"; "XII.6" -> "VI.def.4"; "XII.10" -> "VI.def.4"; "XII.11" -> "VI.def.4"; "VI.2" -> "VI.1"; "VI.15" -> "VI.1"; "VI.19" -> "VI.1"; "VI.20" -> "VI.1"; "VI.4" -> "VI.2"; "VI.11" -> "VI.2"; "VI.12" -> "VI.2"; "XII.3" -> "VI.2"; "VI.6" -> "VI.4"; "VI.18" -> "VI.4"; "VI.20" -> "VI.4"; "XII.1" -> "VI.4"; "VI.20" -> "VI.6"; "XII.1" -> "VI.6"; "VI.19" -> "VI.11"; "VI.22" -> "VI.11"; "VI.22" -> "VI.12"; "VI.19" -> "VI.15"; "VI.22" -> "VI.18"; "VI.20" -> "VI.19"; "VI.22" -> "VI.19"; "XII.1" -> "VI.20"; "XII.4" -> "VI.22"; "VII.def.2" -> "VII.def.1"; "VII.def.15" -> "VII.def.1"; "VII.def.3" -> "VII.def.2"; "VII.def.5" -> "VII.def.2"; "VII.def.15" -> "VII.def.2"; "XI.1" -> "VII.def.3"; "XII.10" -> "VII.def.3"; "X.1" -> "VII.def.5"; "XI.25" -> "VII.def.5"; "X.1" -> "VII.def.6"; "X.1" -> "VII.def.15"; "XII.5" -> "X.1"; "XII.10" -> "X.1"; "XII.11" -> "X.1"; "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.32" -> "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"; "XII.4" -> "XI.def.12"; "XII.5" -> "XI.def.12"; "XII.6" -> "XI.def.12"; "XII.7" -> "XI.def.12"; "XI.28" -> "XI.def.13"; "XI.29" -> "XI.def.13"; "XI.39" -> "XI.def.13"; "XII.3" -> "XI.def.13"; "XII.4" -> "XI.def.13"; "XII.7" -> "XI.def.13"; "XII.10" -> "XI.def.18"; "XII.11" -> "XI.def.18"; "XII.11" -> "XI.def.19"; "XII.10" -> "XI.def.21"; "XII.11" -> "XI.def.21"; "XII.13" -> "XI.def.21"; "XII.13" -> "XI.def.23"; "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.32" -> "XI.25"; "XI.39" -> "XI.28"; "XII.10" -> "XI.28"; "XI.30" -> "XI.29"; "XI.31" -> "XI.29"; "XI.31" -> "XI.30"; "XI.32" -> "XI.31"; "XI.39" -> "XI.31"; "XII.10" -> "XI.32"; "XII.3" -> "XI.39"; "XII.4" -> "XI.39"; "XII.2" -> "XII.1"; "XII.11" -> "XII.1"; "XII.5" -> "XII.2"; "XII.10" -> "XII.2"; "XII.11" -> "XII.2"; "XII.4" -> "XII.3"; "XII.5" -> "XII.3"; "XII.5" -> "XII.4"; "XII.6" -> "XII.5"; "XII.7" -> "XII.5"; "XII.10" -> "XII.6"; "XII.11" -> "XII.6"; "XII.10" -> "XII.7"; "XII.11" -> "XII.10"; "XII.13" -> "XII.11"; }

Required for

XII.14, XII.15

See Commentary on Proposition XII.13.