PROPOSITION 18.

To set out the sides of the five figures and to compare them with one another.

Let AB, the diameter of the given sphere, be set out, and let it be cut at C so that AC is equal to CB, and at D so that AD is double of DB; let the semicircle AEB be described on AB, from C, D let CE, DF be drawn at right angles to AB, and let AF, FB, EB be joined.

Then, since AD is double of DB, therefore AB is triple of BD.
:term:`Convertendo <convertendo>`, therefore, BA is one and a half times AD.
A K C D L B G H E F M N
But, as BA is to AD, so is the square on BA to the square on AF, [v. Def. 9, VI. 8] for the triangle AFB is equiangular with the triangle AFD;
therefore the square on BA is one and a half times the square on AF.
But the square on the diameter of the sphere is also one and a half times the square on the side of the pyramid. [XIII. 13]
And AB is the diameter of the sphere;
therefore AF is equal to the side of the pyramid.
Again, since AD is double of DB,
therefore AB is triple of BD.
But, as AB is to BD, so is the square on AB to the square on BF; [VI. 8, v. Def. 9]
therefore the square on AB is triple of the square on BF.
But the square on the diameter of the sphere is also triple of the square on the side of the cube. [XIII. 15]
And AB is the diameter of the sphere;
therefore BF is the side of the cube.
And, since AC is equal to CB,
therefore AB is double of BC.
But, as AB is to BC, so is the square on AB to the square on BE;
therefore the square on AB is double of the square on BE.
But the square on the diameter of the sphere is also double of the square on the side of the octahedron. [XIII. 14]
And AB is the diameter of the given sphere;
therefore BE is the side of the octahedron.

Next, let AG be drawn from the point A at right angles to the straight line AB, let AG be made equal to AB, let GC be joined, and from H let HK be drawn perpendicular to AB.

Then, since GA is double of AC, for GA is equal to AB, and, as GA is to AC, so is HK to KC,
therefore HK is also double of KC.
Therefore the square on HK is quadruple of the square on KC; therefore the squares on HK, KC, that is, the square on HC, is five times the square on KC.
But HC is equal to CB;
therefore the square on BC is five times the square on CK.
And, since AB is double of CB, and, in them, AD is double of DB,
therefore the remainder BD is double of the remainder DC.
Therefore BC is triple of CD; therefore the square on BC is nine times the square on CD.
But the square on BC is five times the square on CK;
therefore the square on CK is greater than the square on CD; therefore CK is greater than CD.

Let CL be made equal to CK, from L let LM be drawn at right angles to AB, and let MB be joined.

Now, since the square on BC is five times the square on CK, and AB is double of BC, and KL double of CK,
therefore the square on AB is five times the square on KL.
But the square on the diameter of the sphere is also five times the square on the radius of the circle from which the icosahedron has been described. [XIII. 16, Por.]
And AB is the diameter of the sphere;
therefore KL is the radius of the circle from which the icosahedron has been described; therefore KL is a side of the hexagon in the said circle. [IV. 15, Por.]
And, since the diameter of the sphere is made up of the side of the hexagon and two of the sides of the decagon inscribed in the same circle, [XIII. 16, Por.] and AB is the diameter of the sphere, while KL is a side of the hexagon, and AK is equal to LB,
therefore each of the straight lines AK, LB is a side of the decagon inscribed in the circle from which the icosahedron has been described.
And, since LB belongs to a decagon, and ML to a hexagon, for ML is equal to KL, since it is also equal to HK, being the same distance from the centre, and each of the straight lines HK, KL is double of KC,
[XIII. 10]therefore MB belongs to a pentagon.
But the side of the pentagon is the side of the icosahedron ; [XIII. 16]
therefore MB belongs to the icosahedron.
Now, since FB is a side of the cube, let it be cut in extreme and mean ratio at N, and let NB be the greater segment ;
[XIII. 17, Por.]therefore NB is a side of the dodecahedron.
And, since the square on the diameter of the sphere was proved to be one and a half times the square on the side AF of the pyramid, double of the square on the side BE of the octahedron and triple of the side FB of the cube,
therefore, of parts of which the square on the diameter of the sphere contains six, the square on the side of the pyramid contains four, the square on the side of the octahedron three, and the square on the side of the cube two.
Therefore the square on the side of the pyramid is four-thirds of the square on the side of the octahedron, and double of the square on the side of the cube ; and the square on the side of the octahedron is one and a half times the square on the side of the cube.
The said sides, therefore, of the three figures, I mean the pyramid, the octahedron and the cube, are to one another in rational ratios.
But the remaining two, I mean the side of the icosahedron and the side of the dodecahedron, are not in rational ratios either to one another or to the aforesaid sides ; for they are irrational, the one being minor [XIII. 16] and the other an apotome [XIII. 17].
That the side MB of the icosahedron is greater than the side NB of the dodecahedron we can prove thus.
For, since the triangle FDB is equiangular with the triangle FAB, [VI. 8]
[VI. 4]proportionally, as DB is to BF, so is BF to BA.
And, since the three straight lines are proportional, as the first is to the third, so is the square on the first to the square on the second ; [v. Def. 9, VI. 20, Por.]
therefore, as DB is to BA, so is the square on DB to the square on BF;
therefore, inversely, as AB is to BD, so is the square on FB to the square on BD.
But AB is triple of BD ;
therefore the square on FB is triple of the square on BD.
But the square on AD is also quadruple of the square on DB, for AD is double of DB ;
therefore the square on AD is greater than the square on FB ; therefore AD is greater than FB ; therefore AL is by far greater than FB.
And, when AL is cut in extreme and mean ratio, KL is the greater segment, inasmuch as LK belongs to a hexagon, and KA to a decagon ; [XIII. 9] and, when FB is cut in extreme and mean ratio, NB is the greater segment ;
therefore KL is greater than NB.
But KL is equal to LM ;
therefore LM is greater than NB.
Therefore MB, which is a side of the icosahedron, is by far greater than NB which is a side of the dodecahedron.
Q. E. D.

Dependency Graph

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And the centre of the semicircle is the same as that of the circle.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-18", target="_top"]; "I.def.19" [label="I.def.19", fillcolor="#224422", tooltip="I.def.19: Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-19", target="_top"]; "I.def.20" [label="I.def.20", fillcolor="#224422", tooltip="I.def.20: Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-20", target="_top"]; "I.def.21" [label="I.def.21", fillcolor="#224422", tooltip="I.def.21: Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-21", target="_top"]; "I.def.22" [label="I.def.22", fillcolor="#224422", tooltip="I.def.22: Of quadrilateral figures, a square is that which is both equilateral and right-angled ; an oblong that which is right-angled but not equilateral ; a rhombus that which is…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-22", target="_top"]; "I.def.23" [label="I.def.23", fillcolor="#224422", tooltip="I.def.23: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-23", target="_top"]; "I.post.1" [label="I.post.1", fillcolor="#444422", tooltip="I.post.1: To draw a straight line from any point to any point.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-1", target="_top"]; "I.post.2" [label="I.post.2", fillcolor="#444422", tooltip="I.post.2: To produce a finite straight line continuously in a straight line.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-2", target="_top"]; "I.post.3" [label="I.post.3", fillcolor="#444422", tooltip="I.post.3: To describe a circle with any centre and distance.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-3", target="_top"]; "I.post.4" [label="I.post.4", fillcolor="#444422", tooltip="I.post.4: That all right angles are equal to one another.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-4", target="_top"]; "I.post.5" [label="I.post.5", fillcolor="#444422", tooltip="I.post.5: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet…", URL="../../../../volume_I/book_i/postulates/index.html#i-post-5", target="_top"]; "I.cn.1" [label="I.cn.1", fillcolor="#442222", tooltip="I.cn.1: Things which are equal to the same thing are also equal to one another.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-1", target="_top"]; "I.cn.2" [label="I.cn.2", fillcolor="#442222", tooltip="I.cn.2: If equals be added to equals, the wholes are equal.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-2", target="_top"]; "I.cn.3" [label="I.cn.3", fillcolor="#442222", tooltip="I.cn.3: If equals be subtracted from equals, the remainders are equal.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-3", target="_top"]; "I.cn.4" [label="I.cn.4", fillcolor="#442222", tooltip="I.cn.4: Things which coincide with one another are equal to one another.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-4", target="_top"]; "I.cn.5" [label="I.cn.5", fillcolor="#442222", tooltip="I.cn.5: The whole is greater than the part.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-5", target="_top"]; "I.1" [label="I.1", fillcolor="#222244", tooltip="I.1: On a given finite straight line to construct an equilateral triangle.", URL="../../../../volume_I/book_i/propositions/01/index.html", target="_top"]; "I.2" [label="I.2", fillcolor="#222244", tooltip="I.2: To place at a given point (as an extremity) a straight line equal to a given straight line.", URL="../../../../volume_I/book_i/propositions/02/index.html", target="_top"]; "I.3" [label="I.3", fillcolor="#222244", tooltip="I.3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less.", URL="../../../../volume_I/book_i/propositions/03/index.html", target="_top"]; "I.4" [label="I.4", fillcolor="#222244", tooltip="I.4: If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the…", URL="../../../../volume_I/book_i/propositions/04/index.html", target="_top"]; "I.5" [label="I.5", fillcolor="#222244", tooltip="I.5: In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one…", URL="../../../../volume_I/book_i/propositions/05/index.html", target="_top"]; "I.6" [label="I.6", fillcolor="#222244", tooltip="I.6: If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another.", URL="../../../../volume_I/book_i/propositions/06/index.html", target="_top"]; "I.7" [label="I.7", fillcolor="#222244", tooltip="I.7: Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be constructed on the same straight line (from its…", URL="../../../../volume_I/book_i/propositions/07/index.html", target="_top"]; "I.8" [label="I.8", fillcolor="#222244", tooltip="I.8: If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the…", URL="../../../../volume_I/book_i/propositions/08/index.html", target="_top"]; "I.9" [label="I.9", fillcolor="#222244", tooltip="I.9: To bisect a given rectilineal angle.", URL="../../../../volume_I/book_i/propositions/09/index.html", target="_top"]; "I.10" [label="I.10", fillcolor="#222244", tooltip="I.10: To bisect a given finite straight line.", URL="../../../../volume_I/book_i/propositions/10/index.html", target="_top"]; "I.11" [label="I.11", fillcolor="#222244", tooltip="I.11: To draw a straight line at right angles to a given straight line from a given point on it.", URL="../../../../volume_I/book_i/propositions/11/index.html", target="_top"]; "I.12" [label="I.12", fillcolor="#222244", tooltip="I.12: To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line.", URL="../../../../volume_I/book_i/propositions/12/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"]; "I.46" [label="I.46", fillcolor="#222244", tooltip="I.46: On a given straight line to describe a square.", URL="../../../../volume_I/book_i/propositions/46/index.html", target="_top"]; "I.47" [label="I.47", fillcolor="#222244", tooltip="I.47: In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.", URL="../../../../volume_I/book_i/propositions/47/index.html", target="_top"]; "II.def.1" [label="II.def.1", fillcolor="#224422", tooltip="II.def.1: Any rectangular parallelogram is said to be contained by the two straight lines containing the right angle.", URL="../../../../volume_I/book_ii/definitions/index.html#ii-def-1", target="_top"]; "II.def.2" [label="II.def.2", fillcolor="#224422", tooltip="II.def.2: And in any parallelogrammic area let any one whatever of the parallelograms about its diameter with the two complements be called a gnomon.", URL="../../../../volume_I/book_ii/definitions/index.html#ii-def-2", target="_top"]; "II.2" [label="II.2", fillcolor="#222244", tooltip="II.2: If a straight line be cut at random, the rectangle contained by the whole and both of the segments is equal to the square on the whole.", URL="../../../../volume_I/book_ii/propositions/02/index.html", target="_top"]; "II.5" [label="II.5", fillcolor="#222244", tooltip="II.5: If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the…", URL="../../../../volume_I/book_ii/propositions/05/index.html", target="_top"]; "II.7" [label="II.7", fillcolor="#222244", tooltip="II.7: If a straight line be cut at random, the square on the whole and that on one of the segments both together are equal to twice the rectangle contained by the whole and the said…", URL="../../../../volume_I/book_ii/propositions/07/index.html", 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.19" [label="III.19", fillcolor="#222244", tooltip="III.19: If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the tangent, the centre of the circle will be on the straight line so…", URL="../../../../volume_II/book_iii/propositions/19/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.28" [label="III.28", fillcolor="#222244", tooltip="III.28: In equal circles equal straight lines cut off equal circumferences, the greater equal to the greater and the less to the less.", URL="../../../../volume_II/book_iii/propositions/28/index.html", target="_top"]; "III.29" [label="III.29", fillcolor="#222244", tooltip="III.29: In equal circles equal circumferences are subtended by equal straight lines.", URL="../../../../volume_II/book_iii/propositions/29/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"]; "III.32" [label="III.32", fillcolor="#222244", tooltip="III.32: If a straight line touch a circle, and from the point of contact there be drawn across, in the circle, a straight line cutting the circle, the angles which it makes with the…", URL="../../../../volume_II/book_iii/propositions/32/index.html", target="_top"]; "IV.def.2" [label="IV.def.2", fillcolor="#224422", tooltip="IV.def.2: Similarly a figure is said to be circumscribed about a figure when the respective sides of the circumscribed figure pass through the respective angles of that about which it is…", URL="../../../../volume_II/book_iv/definitions/index.html#iv-def-2", 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.def.5" [label="IV.def.5", fillcolor="#224422", tooltip="IV.def.5: Similarly a circle is said to be inscribed in a figure when the circumference of the circle touches each side of the figure in which it is inscribed.", URL="../../../../volume_II/book_iv/definitions/index.html#iv-def-5", target="_top"]; "IV.def.6" [label="IV.def.6", fillcolor="#224422", tooltip="IV.def.6: A circle is said to be circumscribed about a figure when the circumference of the circle passes through each angle of the figure about which it is circumscribed.", URL="../../../../volume_II/book_iv/definitions/index.html#iv-def-6", target="_top"]; "IV.def.7" [label="IV.def.7", fillcolor="#224422", tooltip="IV.def.7: A straight line is said to be fitted into a circle when its extremities are on the circumference of the circle.", URL="../../../../volume_II/book_iv/definitions/index.html#iv-def-7", target="_top"]; "IV.1" [label="IV.1", fillcolor="#222244", tooltip="IV.1: Into a given circle to fit a straight line equal to a given straight line which is not greater than the diameter of the circle.", URL="../../../../volume_II/book_iv/propositions/01/index.html", target="_top"]; "IV.2" [label="IV.2", fillcolor="#222244", tooltip="IV.2: In a given circle to inscribe a triangle equiangular with a given triangle.", URL="../../../../volume_II/book_iv/propositions/02/index.html", target="_top"]; "IV.14" [label="IV.14", fillcolor="#222244", tooltip="IV.14: About a given pentagon, which is equilateral and equiangular, to circumscribe a circle.", URL="../../../../volume_II/book_iv/propositions/14/index.html", target="_top"]; "IV.15" [label="IV.15", fillcolor="#222244", tooltip="IV.15: In a given circle to inscribe an equilateral and equiangular hexagon.", URL="../../../../volume_II/book_iv/propositions/15/index.html", target="_top"]; "V.def.1" [label="V.def.1", fillcolor="#224422", tooltip="V.def.1: A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-1", target="_top"]; "V.def.2" [label="V.def.2", fillcolor="#224422", tooltip="V.def.2: The greater is a multiple of the less when it is measured by the less.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-2", target="_top"]; "V.def.3" [label="V.def.3", fillcolor="#224422", tooltip="V.def.3: A ratio is a sort of relation in respect of size between two magnitudes of the same kind.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-3", target="_top"]; "V.def.4" [label="V.def.4", fillcolor="#224422", tooltip="V.def.4: Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-4", target="_top"]; "V.def.5" [label="V.def.5", fillcolor="#224422", tooltip="V.def.5: Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any…", URL="../../../../volume_II/book_v/definitions/index.html#v-def-5", target="_top"]; "V.def.6" [label="V.def.6", fillcolor="#224422", tooltip="V.def.6: Let magnitudes which have the same ratio be called proportional.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-6", target="_top"]; "V.def.7" [label="V.def.7", fillcolor="#224422", tooltip="V.def.7: When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then…", URL="../../../../volume_II/book_v/definitions/index.html#v-def-7", target="_top"]; "V.def.9" [label="V.def.9", fillcolor="#224422", tooltip="V.def.9: When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-9", target="_top"]; "V.def.11" [label="V.def.11", fillcolor="#224422", tooltip="V.def.11: The term corresponding magnitudes is used of antecedents in relation to antecedents, and of consequents in relation to consequents.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-11", target="_top"]; "V.def.12" [label="V.def.12", fillcolor="#224422", tooltip="V.def.12: Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-12", target="_top"]; "V.def.13" [label="V.def.13", fillcolor="#224422", tooltip="V.def.13: Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-13", target="_top"]; "V.def.14" [label="V.def.14", fillcolor="#224422", tooltip="V.def.14: Composition of a ratio means taking the antecedent together with the consequent as one in relation to the consequent by itself.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-14", target="_top"]; "V.def.15" [label="V.def.15", fillcolor="#224422", tooltip="V.def.15: Separation of a ratio means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-15", target="_top"]; "V.def.16" [label="V.def.16", fillcolor="#224422", tooltip="V.def.16: Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.", URL="../../../../volume_II/book_v/definitions/index.html#v-def-16", target="_top"]; "V.def.17" [label="V.def.17", fillcolor="#224422", tooltip="V.def.17: A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, as the first is to…", URL="../../../../volume_II/book_v/definitions/index.html#v-def-17", target="_top"]; "V.1" [label="V.1", fillcolor="#222244", tooltip="V.1: If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple one of the magnitudes is of one…", URL="../../../../volume_II/book_v/propositions/01/index.html", target="_top"]; "V.2" [label="V.2", fillcolor="#222244", tooltip="V.2: If a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same multiple of the second that a sixth is of the fourth, the sum of…", URL="../../../../volume_II/book_v/propositions/02/index.html", target="_top"]; "V.3" [label="V.3", fillcolor="#222244", tooltip="V.3: If a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the first and third, then also ex aequali the magnitudes taken…", URL="../../../../volume_II/book_v/propositions/03/index.html", target="_top"]; "V.4" [label="V.4", fillcolor="#222244", tooltip="V.4: If a first magnitude have to a second the same ratio as a third to a fourth, any equimultiples whatever of the first and third will also have the same ratio to any equimultiples…", URL="../../../../volume_II/book_v/propositions/04/index.html", target="_top"]; "V.7" [label="V.7", fillcolor="#222244", tooltip="V.7: Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.", URL="../../../../volume_II/book_v/propositions/07/index.html", target="_top"]; "V.8" [label="V.8", fillcolor="#222244", tooltip="V.8: Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.", URL="../../../../volume_II/book_v/propositions/08/index.html", target="_top"]; "V.9" [label="V.9", fillcolor="#222244", tooltip="V.9: Magnitudes which have the same ratio to the same are equal to one another ; and magnitudes to which the same has the same ratio are equal.", URL="../../../../volume_II/book_v/propositions/09/index.html", target="_top"]; "V.10" [label="V.10", fillcolor="#222244", tooltip="V.10: Of magnitudes which have a ratio to the same, that which has a greater ratio is greater ; and that to which the same has a greater ratio is less.", URL="../../../../volume_II/book_v/propositions/10/index.html", target="_top"]; "V.11" [label="V.11", fillcolor="#222244", tooltip="V.11: Ratios which are the same with the same ratio are also the same with one another.", URL="../../../../volume_II/book_v/propositions/11/index.html", target="_top"]; "V.12" [label="V.12", fillcolor="#222244", tooltip="V.12: If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.", URL="../../../../volume_II/book_v/propositions/12/index.html", target="_top"]; "V.13" [label="V.13", fillcolor="#222244", tooltip="V.13: If a first magnitude have to a second the same ratio as a third to a fourth, and the third have to the fourth a greater ratio than a fifth has to a sixth, the first will also have…", URL="../../../../volume_II/book_v/propositions/13/index.html", target="_top"]; "V.14" [label="V.14", fillcolor="#222244", tooltip="V.14: If a first magnitude have to a second the same ratio as a third has to a fourth, and the first be greater than the third, the second will also be greater than the fourth ; if…", URL="../../../../volume_II/book_v/propositions/14/index.html", target="_top"]; "V.15" [label="V.15", fillcolor="#222244", tooltip="V.15: Parts have the same ratio as the same multiples of them taken in corresponding order.", URL="../../../../volume_II/book_v/propositions/15/index.html", target="_top"]; "V.16" [label="V.16", fillcolor="#222244", tooltip="V.16: If four magnitudes be proportional, they will also be proportional alternately.", URL="../../../../volume_II/book_v/propositions/16/index.html", target="_top"]; "V.17" [label="V.17", fillcolor="#222244", tooltip="V.17: If magnitudes be proportional componendo, they will also be proportional separando.", URL="../../../../volume_II/book_v/propositions/17/index.html", target="_top"]; "V.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.19" [label="V.19", fillcolor="#222244", tooltip="V.19: If, as a whole is to a whole, so is a part subtracted to a part subtracted, the remainder will also be to the remainder as whole to whole.", URL="../../../../volume_II/book_v/propositions/19/index.html", target="_top"]; "V.20" [label="V.20", fillcolor="#222244", tooltip="V.20: If there be three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first be greater than the third, the…", URL="../../../../volume_II/book_v/propositions/20/index.html", target="_top"]; "V.22" [label="V.22", fillcolor="#222244", tooltip="V.22: If there be any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, they will also be in the same ratio…", URL="../../../../volume_II/book_v/propositions/22/index.html", target="_top"]; "VI.def.1" [label="VI.def.1", fillcolor="#224422", tooltip="VI.def.1: Similar rectilineal figures are such as have their angles severally equal and the sides about the equal angles proportional.", URL="../../../../volume_II/book_vi/definitions/index.html#vi-def-1", target="_top"]; "VI.def.2" [label="VI.def.2", fillcolor="#224422", tooltip="VI.def.2: [2. Reciprocally related figures. See note.]", URL="../../../../volume_II/book_vi/definitions/index.html#vi-def-2", target="_top"]; "VI.def.3" [label="VI.def.3", fillcolor="#224422", tooltip="VI.def.3: A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.", URL="../../../../volume_II/book_vi/definitions/index.html#vi-def-3", 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.8" [label="VI.8", fillcolor="#222244", tooltip="VI.8: If in a right-angled triangle a perpendicular be drawn from the right angle to the base, the triangles adjoining the perpendicular are similar both to the whole and to one another.", URL="../../../../volume_II/book_vi/propositions/08/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.13" [label="VI.13", fillcolor="#222244", tooltip="VI.13: To two given straight lines to find a mean proportional.", URL="../../../../volume_II/book_vi/propositions/13/index.html", target="_top"]; "VI.14" [label="VI.14", fillcolor="#222244", tooltip="VI.14: In equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; and equiangular parallelograms in which the sides about the equal angles…", URL="../../../../volume_II/book_vi/propositions/14/index.html", target="_top"]; "VI.15" [label="VI.15", fillcolor="#222244", tooltip="VI.15: In equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally proportional; and those triangles which have one angle equal to one…", URL="../../../../volume_II/book_vi/propositions/15/index.html", target="_top"]; "VI.16" [label="VI.16", fillcolor="#222244", tooltip="VI.16: If four straight lines be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means; and, if the rectangle contained by the extremes…", URL="../../../../volume_II/book_vi/propositions/16/index.html", target="_top"]; "VI.17" [label="VI.17", fillcolor="#222244", tooltip="VI.17: If three straight lines be proportional, the rectangle contained by the extremes is equal to the square on the mean; and, if the rectangle contained by the extremes be equal to…", URL="../../../../volume_II/book_vi/propositions/17/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.21" [label="VI.21", fillcolor="#222244", tooltip="VI.21: Figures which are similar to the same rectilineal figure are also similar to one another.", URL="../../../../volume_II/book_vi/propositions/21/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"]; "VI.24" [label="VI.24", fillcolor="#222244", tooltip="VI.24: In any parallelogram the parallelograms about the diameter are similar both to the whole and to one another.", URL="../../../../volume_II/book_vi/propositions/24/index.html", target="_top"]; "VI.25" [label="VI.25", fillcolor="#222244", tooltip="VI.25: To construct one and the same figure similar to a given rectilineal figure and equal to another given rectilineal figure.", URL="../../../../volume_II/book_vi/propositions/25/index.html", target="_top"]; "VI.26" [label="VI.26", fillcolor="#222244", tooltip="VI.26: If from a parallelogram there be taken away a parallelogram similar and similarly situated to the whole and having a common angle with it, it is about the same diameter with the…", URL="../../../../volume_II/book_vi/propositions/26/index.html", target="_top"]; "VI.28" [label="VI.28", fillcolor="#222244", tooltip="VI.28: To a given straight line to apply a parallelogram equal to a given rectilineal figure and deficient by a parallelogrammic figure similar to a given one : thus the given…", URL="../../../../volume_II/book_vi/propositions/28/index.html", target="_top"]; "VI.32" [label="VI.32", fillcolor="#222244", tooltip="VI.32: If two triangles having two sides proportional to two sides be placed together at one angle so that their corresponding sides are also parallel, the remaining sides of the…", URL="../../../../volume_II/book_vi/propositions/32/index.html", target="_top"]; "VI.33" [label="VI.33", fillcolor="#222244", tooltip="VI.33: In equal circles angles have the same ratio as the circumferences on which they stand, whether they stand at the centres or at the circumferences.", URL="../../../../volume_II/book_vi/propositions/33/index.html", target="_top"]; "VII.def.1" [label="VII.def.1", fillcolor="#224422", tooltip="VII.def.1: An unit is that by virtue of which each of the things that exist is called one.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-1", target="_top"]; "VII.def.2" [label="VII.def.2", fillcolor="#224422", tooltip="VII.def.2: A number is a multitude composed of units.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-2", target="_top"]; "VII.def.3" [label="VII.def.3", fillcolor="#224422", tooltip="VII.def.3: A number is a part of a number, the less of the greater, when it measures the greater.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-3", target="_top"]; "VII.def.4" [label="VII.def.4", fillcolor="#224422", tooltip="VII.def.4: but parts when it does not measure it.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-4", target="_top"]; "VII.def.5" [label="VII.def.5", fillcolor="#224422", tooltip="VII.def.5: The greater number is a multiple of the less when it is measured by the less.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-5", target="_top"]; "VII.def.11" [label="VII.def.11", fillcolor="#224422", tooltip="VII.def.11: A prime number is that which is measured by an unit alone.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-11", target="_top"]; "VII.def.12" [label="VII.def.12", fillcolor="#224422", tooltip="VII.def.12: Numbers prime to one another are those which are measured by an unit alone as a common measure.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-12", target="_top"]; "VII.def.15" [label="VII.def.15", fillcolor="#224422", tooltip="VII.def.15: A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-15", target="_top"]; "VII.def.16" [label="VII.def.16", fillcolor="#224422", tooltip="VII.def.16: And, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-16", target="_top"]; "VII.def.18" [label="VII.def.18", fillcolor="#224422", tooltip="VII.def.18: A square number is equal multiplied by equal, or a number which is contained by two equal numbers.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-18", target="_top"]; "VII.def.20" [label="VII.def.20", fillcolor="#224422", tooltip="VII.def.20: Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-20", target="_top"]; "VII.def.21" [label="VII.def.21", fillcolor="#224422", tooltip="VII.def.21: Similar plane and solid numbers are those which have their sides proportional.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-21", target="_top"]; "VII.1" [label="VII.1", fillcolor="#222244", tooltip="VII.1: Two unequal numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit…", URL="../../../../volume_II/book_vii/propositions/01/index.html", target="_top"]; "VII.2" [label="VII.2", fillcolor="#222244", tooltip="VII.2: Given two numbers not prime to one another, to find their greatest common measure.", URL="../../../../volume_II/book_vii/propositions/02/index.html", target="_top"]; "VII.3" [label="VII.3", fillcolor="#222244", tooltip="VII.3: Given three numbers not prime to one another, to find their greatest common measure.", URL="../../../../volume_II/book_vii/propositions/03/index.html", target="_top"]; "VII.4" [label="VII.4", fillcolor="#222244", tooltip="VII.4: Any number is either a part or parts of any number, the less of the greater.", URL="../../../../volume_II/book_vii/propositions/04/index.html", target="_top"]; "VII.5" [label="VII.5", fillcolor="#222244", tooltip="VII.5: If a number be a part of a number, and another be the same part of another, the sum will also be the same part of the sum that the one is of the one.", URL="../../../../volume_II/book_vii/propositions/05/index.html", target="_top"]; "VII.6" [label="VII.6", fillcolor="#222244", tooltip="VII.6: If a number be parts of a number, and another be the same parts of another, the sum will also be the same parts of the sum that the one is of the one.", URL="../../../../volume_II/book_vii/propositions/06/index.html", target="_top"]; "VII.9" [label="VII.9", fillcolor="#222244", tooltip="VII.9: If a number be a part of a number, and another be the same part of another, alternately also, whatever part or parts the first is of the third, the same part, or the same parts…", URL="../../../../volume_II/book_vii/propositions/09/index.html", target="_top"]; "VII.10" [label="VII.10", fillcolor="#222244", tooltip="VII.10: If a number be parts of a number, and another be the same parts of another, alternately also, whatever parts or part the first is of the third, the same parts or the same part…", URL="../../../../volume_II/book_vii/propositions/10/index.html", target="_top"]; "VII.12" [label="VII.12", fillcolor="#222244", tooltip="VII.12: If there be as many numbers as we please in proportion, then, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents.", URL="../../../../volume_II/book_vii/propositions/12/index.html", target="_top"]; "VII.13" [label="VII.13", fillcolor="#222244", tooltip="VII.13: If four numbers be proportional, they will also be proportional alternately.", URL="../../../../volume_II/book_vii/propositions/13/index.html", target="_top"]; "VII.14" [label="VII.14", fillcolor="#222244", tooltip="VII.14: If there be as many numbers as we please, and others equal to them in multitude, which taken two and two are in the same ratio, they will also be in the same ratio ex aequali.", URL="../../../../volume_II/book_vii/propositions/14/index.html", target="_top"]; "VII.15" [label="VII.15", fillcolor="#222244", tooltip="VII.15: If an unit measure any number, and another number measure any other number the same number of times, alternately also, the unit will measure the third number the same number of…", URL="../../../../volume_II/book_vii/propositions/15/index.html", target="_top"]; "VII.16" [label="VII.16", fillcolor="#222244", tooltip="VII.16: If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.", URL="../../../../volume_II/book_vii/propositions/16/index.html", target="_top"]; "VII.17" [label="VII.17", fillcolor="#222244", tooltip="VII.17: If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.", URL="../../../../volume_II/book_vii/propositions/17/index.html", target="_top"]; "VII.18" [label="VII.18", fillcolor="#222244", tooltip="VII.18: If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.", URL="../../../../volume_II/book_vii/propositions/18/index.html", target="_top"]; "VII.19" [label="VII.19", fillcolor="#222244", tooltip="VII.19: If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the…", URL="../../../../volume_II/book_vii/propositions/19/index.html", target="_top"]; "VII.20" [label="VII.20", fillcolor="#222244", tooltip="VII.20: The least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less.", URL="../../../../volume_II/book_vii/propositions/20/index.html", target="_top"]; "VII.21" [label="VII.21", fillcolor="#222244", tooltip="VII.21: Numbers prime to one another are the least of those which have the same ratio with them.", URL="../../../../volume_II/book_vii/propositions/21/index.html", target="_top"]; "VII.22" [label="VII.22", fillcolor="#222244", tooltip="VII.22: The least numbers of those which have the same ratio with them are prime to one another.", URL="../../../../volume_II/book_vii/propositions/22/index.html", target="_top"]; "VII.23" [label="VII.23", fillcolor="#222244", tooltip="VII.23: If two numbers be prime to one another, the number which measures the one of them will be prime to the remaining number.", URL="../../../../volume_II/book_vii/propositions/23/index.html", target="_top"]; "VII.24" [label="VII.24", fillcolor="#222244", tooltip="VII.24: If two numbers be prime to any number, their product also will be prime to the same.", URL="../../../../volume_II/book_vii/propositions/24/index.html", target="_top"]; "VII.25" [label="VII.25", fillcolor="#222244", tooltip="VII.25: If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.", URL="../../../../volume_II/book_vii/propositions/25/index.html", target="_top"]; "VII.26" [label="VII.26", fillcolor="#222244", tooltip="VII.26: If two numbers be prime to two numbers, both to each, their products also will be prime to one another.", URL="../../../../volume_II/book_vii/propositions/26/index.html", target="_top"]; "VII.27" [label="VII.27", fillcolor="#222244", tooltip="VII.27: If two numbers be prime to one another, and each by multiplying itself make a certain number, the products will be prime to one another ; and, if the original numbers by…", URL="../../../../volume_II/book_vii/propositions/27/index.html", target="_top"]; "VII.33" [label="VII.33", fillcolor="#222244", tooltip="VII.33: Given as many numbers as we please, to find the least of those which have the same ratio with them.", URL="../../../../volume_II/book_vii/propositions/33/index.html", target="_top"]; "VII.34" [label="VII.34", fillcolor="#222244", tooltip="VII.34: Given two numbers, to find the least number which they measure.", URL="../../../../volume_II/book_vii/propositions/34/index.html", target="_top"]; "VII.35" [label="VII.35", fillcolor="#222244", tooltip="VII.35: If two numbers measure any number, the least number measured by them will also measure the same.", URL="../../../../volume_II/book_vii/propositions/35/index.html", target="_top"]; "VIII.def.20" [label="VIII.def.20", fillcolor="#224422", tooltip="VIII.def.20: Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-20", target="_top"]; "VIII.1" [label="VIII.1", fillcolor="#222244", tooltip="VIII.1: If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the numbers are the least of those which have the same ratio…", URL="../../../../volume_II/book_viii/propositions/01/index.html", target="_top"]; "VIII.2" [label="VIII.2", fillcolor="#222244", tooltip="VIII.2: To find numbers in continued proportion, as many as may be prescribed, and the least that are in a given ratio.", URL="../../../../volume_II/book_viii/propositions/02/index.html", target="_top"]; "VIII.4" [label="VIII.4", fillcolor="#222244", tooltip="VIII.4: Given as many ratios as we please in least numbers, to find numbers in continued proportion which are the least in the given ratios.", URL="../../../../volume_II/book_viii/propositions/04/index.html", target="_top"]; "VIII.11" [label="VIII.11", fillcolor="#222244", tooltip="VIII.11: Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate of that which the side has to the side.", URL="../../../../volume_II/book_viii/propositions/11/index.html", target="_top"]; "VIII.18" [label="VIII.18", fillcolor="#222244", tooltip="VIII.18: Between two similar plane numbers there is one mean proportional number; and the plane number has to the plane number the ratio duplicate of that which the corresponding side has…", URL="../../../../volume_II/book_viii/propositions/18/index.html", target="_top"]; "VIII.26" [label="VIII.26", fillcolor="#222244", tooltip="VIII.26: Similar plane numbers have to one another the ratio which a square number has to a square number.", URL="../../../../volume_II/book_viii/propositions/26/index.html", target="_top"]; "X.def.1" [label="X.def.1", fillcolor="#224422", tooltip="X.def.1: Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.", URL="../../../book_x/definitions/index.html#x-def-1", target="_top"]; "X.def.2" [label="X.def.2", fillcolor="#224422", tooltip="X.def.2: Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when the squares on them cannot possibly have any…", URL="../../../book_x/definitions/index.html#x-def-2", target="_top"]; "X.def.3" [label="X.def.3", fillcolor="#224422", tooltip="X.def.3: With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others…", URL="../../../book_x/definitions/index.html#x-def-3", target="_top"]; "X.def.4" [label="X.def.4", fillcolor="#224422", tooltip="X.def.4: And let the square on the assigned straight line be called rational and those areas which are commensurable with it rational, but those which are incommensurable with it…", URL="../../../book_x/definitions/index.html#x-def-4", target="_top"]; "X.def.iii.1" [label="X.def.iii.1", fillcolor="#224422", tooltip="X.def.iii.1: Given a rational straight line and an apotome, if the square on the whole be greater than the square on the annex by the square on a straight line commensurable in length with the…", URL="../../../book_x/definitions_iii/index.html#x-def-iii-1", target="_top"]; "X.def.iii.4" [label="X.def.iii.4", fillcolor="#224422", tooltip="X.def.iii.4: Again, if the square on the whole be greater than the square on the annex by the square on a straight line incommensurable with the whole, then, if the whole be commensurable in…", URL="../../../book_x/definitions_iii/index.html#x-def-iii-4", target="_top"]; "X.5" [label="X.5", fillcolor="#222244", tooltip="X.5: Commensurable magnitudes have to one another the ratio which a number has to a number.", URL="../../../book_x/propositions/005/index.html", target="_top"]; "X.6" [label="X.6", fillcolor="#222244", tooltip="X.6: If two magnitudes have to one another the ratio which a number has to a number, the magnitudes will be commensurable.", URL="../../../book_x/propositions/006/index.html", target="_top"]; "X.7" [label="X.7", fillcolor="#222244", tooltip="X.7: Incommensurable magnitudes have not to one another the ratio which a number has to a number.", URL="../../../book_x/propositions/007/index.html", target="_top"]; "X.8" [label="X.8", fillcolor="#222244", tooltip="X.8: If two magnitudes have not to one another the ratio which a number has to a number, the magnitudes will be incommensurable.", URL="../../../book_x/propositions/008/index.html", target="_top"]; "X.9" [label="X.9", fillcolor="#222244", tooltip="X.9: The squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio…", URL="../../../book_x/propositions/009/index.html", target="_top"]; "X.11" [label="X.11", fillcolor="#222244", tooltip="X.11: If four magnitudes be proportional, and the first be commensurable with the second, the third will also be commensurable with the fourth ; and, if the first be incommensurable…", URL="../../../book_x/propositions/011/index.html", target="_top"]; "X.12" [label="X.12", fillcolor="#222244", tooltip="X.12: Magnitudes commensurable with the same magnitude are commensurable with one another also.", URL="../../../book_x/propositions/012/index.html", target="_top"]; "X.13" [label="X.13", fillcolor="#222244", tooltip="X.13: If two magnitudes be commensurable, and the one of them be incommensurable with any magnitude, the remaining one will also be incommensurable with the same.", URL="../../../book_x/propositions/013/index.html", target="_top"]; "X.15" [label="X.15", fillcolor="#222244", tooltip="X.15: If two commensurable magnitudes be added together, the whole will also be commensurable with each of them; and, if the whole be commensurable with one of them, the original…", URL="../../../book_x/propositions/015/index.html", target="_top"]; "X.16" [label="X.16", fillcolor="#222244", tooltip="X.16: If two incommensurable magnitudes be added together, the whole will also be incommensurable with each of them; and, if the whole be incommensurable with one of them, the original…", URL="../../../book_x/propositions/016/index.html", target="_top"]; "X.17" [label="X.17", fillcolor="#222244", tooltip="X.17: If there be two unequal straight lines, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure…", URL="../../../book_x/propositions/017/index.html", target="_top"]; "X.18" [label="X.18", fillcolor="#222244", tooltip="X.18: If there be two unequal straight lines, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure…", URL="../../../book_x/propositions/018/index.html", target="_top"]; "X.19" [label="X.19", fillcolor="#222244", tooltip="X.19: The rectangle contained by rational straight lines commensurable in length is rational.", URL="../../../book_x/propositions/019/index.html", target="_top"]; "X.20" [label="X.20", fillcolor="#222244", tooltip="X.20: If a rational area be applied to a rational straight line, it produces as breadth a straight line rational and commensurable in length with the straight line to which it is…", URL="../../../book_x/propositions/020/index.html", target="_top"]; "X.21" [label="X.21", fillcolor="#222244", tooltip="X.21: The rectangle contained by rational straight lines commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called…", URL="../../../book_x/propositions/021/index.html", target="_top"]; "X.22" [label="X.22", fillcolor="#222244", tooltip="X.22: The square on a medial straight line, if applied to a rational straight line, produces as breadth a straight line rational and incommensurable in length with that to which it is…", URL="../../../book_x/propositions/022/index.html", target="_top"]; "X.23" [label="X.23", fillcolor="#222244", tooltip="X.23: A straight line commensurable with a medial straight line is medial.", URL="../../../book_x/propositions/023/index.html", target="_top"]; "X.30" [label="X.30", fillcolor="#222244", tooltip="X.30: To find two rational straight lines commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a straight line…", URL="../../../book_x/propositions/030/index.html", target="_top"]; "X.33" [label="X.33", fillcolor="#222244", tooltip="X.33: To find two straight lines incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial.", URL="../../../book_x/propositions/033/index.html", target="_top"]; "X.73" [label="X.73", fillcolor="#222244", tooltip="X.73: If from a rational straight line there be subtracted a rational straight line commensurable with the whole in square only, the remainder is irrational ; and let it be called an…", URL="../../../book_x/propositions/073/index.html", target="_top"]; "X.76" [label="X.76", fillcolor="#222244", tooltip="X.76: If from a straight line there be subtracted a straight line which is incommensurable in square with the whole and which with the whole makes the squares on them added together…", URL="../../../book_x/propositions/076/index.html", target="_top"]; "X.94" [label="X.94", fillcolor="#222244", tooltip="X.94: If an area be contained by a rational straight line and a fourth apotome, the “side” of the area is minor.", URL="../../../book_x/propositions/094/index.html", target="_top"]; "X.97" [label="X.97", fillcolor="#222244", tooltip="X.97: The square on an apotome applied to a rational straight line produces as breadth a first apotome.", URL="../../../book_x/propositions/097/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.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.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.14" [label="XI.def.14", fillcolor="#224422", tooltip="XI.def.14: When, the diameter of a semicircle remaining fixed, the semicircle is carried round and restored again to the same position from which it began to be moved, the figure so…", URL="../../../book_xi/definitions/index.html#xi-def-14", target="_top"]; "XI.def.15" [label="XI.def.15", fillcolor="#224422", tooltip="XI.def.15: The axis of the sphere is the straight line which remains fixed and about which the semicircle is turned.", URL="../../../book_xi/definitions/index.html#xi-def-15", target="_top"]; "XI.def.16" [label="XI.def.16", fillcolor="#224422", tooltip="XI.def.16: The centre of the sphere is the same as that of the semicircle.", URL="../../../book_xi/definitions/index.html#xi-def-16", target="_top"]; "XI.def.17" [label="XI.def.17", fillcolor="#224422", tooltip="XI.def.17: A diameter of the sphere is any straight line drawn through the centre and terminated in both directions by the surface of the sphere.", URL="../../../book_xi/definitions/index.html#xi-def-17", target="_top"]; "XI.def.25" [label="XI.def.25", fillcolor="#224422", tooltip="XI.def.25: A cube is a solid figure contained by six equal squares.", URL="../../../book_xi/definitions/index.html#xi-def-25", target="_top"]; "XI.def.26" [label="XI.def.26", fillcolor="#224422", tooltip="XI.def.26: An octahedron is a solid figure contained by eight equal and equilateral triangles.", URL="../../../book_xi/definitions/index.html#xi-def-26", target="_top"]; "XI.def.27" [label="XI.def.27", fillcolor="#224422", tooltip="XI.def.27: An icosahedron is a solid figure contained by twenty equal and equilateral triangles.", URL="../../../book_xi/definitions/index.html#xi-def-27", target="_top"]; "XI.def.28" [label="XI.def.28", fillcolor="#224422", tooltip="XI.def.28: A dodecahedron is a solid figure contained by twelve equal, equilateral, and equiangular pentagons.", URL="../../../book_xi/definitions/index.html#xi-def-28", 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.11" [label="XI.11", fillcolor="#222244", tooltip="XI.11: From a given elevated point to draw a straight line perpendicular to a given plane.", URL="../../../book_xi/propositions/11/index.html", target="_top"]; "XI.12" [label="XI.12", fillcolor="#222244", tooltip="XI.12: To set up a straight line at right angles to a given plane from a given point in it.", URL="../../../book_xi/propositions/12/index.html", target="_top"]; "XI.38" [label="XI.38", fillcolor="#222244", tooltip="XI.38: If the sides of the opposite planes of a cube be bisected, and planes be carried through the points of section, the common section of the planes and the diameter of the cube…", URL="../../../book_xi/propositions/38/index.html", target="_top"]; "XIII.1" [label="XIII.1", fillcolor="#222244", tooltip="XIII.1: If a straight line be cut in extreme and mean ratio, the square on the greater segment added to the half of the whole is five times the square on the half.", URL="../01/index.html", target="_top"]; "XIII.3" [label="XIII.3", fillcolor="#222244", tooltip="XIII.3: If a straight line be cut in extreme and mean ratio, the square on the lesser segment added to the half of the greater segment is five times the square on the half of the greater…", URL="../03/index.html", target="_top"]; "XIII.4" [label="XIII.4", fillcolor="#222244", tooltip="XIII.4: If a straight line be cut in extreme and mean ratio, the square on the whole and the square on the lesser segment together are triple of the square on the greater segment.", URL="../04/index.html", target="_top"]; "XIII.5" [label="XIII.5", fillcolor="#222244", tooltip="XIII.5: If a straight line be cut in extreme and mean ratio, and there be added to it a straight line equal to the greater segment, the whole straight line has been cut in extreme and…", URL="../05/index.html", target="_top"]; "XIII.6" [label="XIII.6", fillcolor="#222244", tooltip="XIII.6: If a rational straight line be cut in extreme and mean ratio, each of the segments is the irrational straight line called apotome.", URL="../06/index.html", target="_top"]; "XIII.7" [label="XIII.7", fillcolor="#222244", tooltip="XIII.7: If three angles of an equilateral pentagon, taken either in order or not in order, be equal, the pentagon will be equiangular.", URL="../07/index.html", target="_top"]; "XIII.8" [label="XIII.8", fillcolor="#222244", tooltip="XIII.8: If in an equilateral and equiangular pentagon straight lines subtend two angles taken in order, they cut one another in extreme and mean ratio, and their greater segments are…", URL="../08/index.html", target="_top"]; "XIII.9" [label="XIII.9", fillcolor="#222244", tooltip="XIII.9: If the side of the hexagon and that of the decagon inscribed in the same circle be added together, the whole straight line has been cut in extreme and mean ratio, and its greater…", URL="../09/index.html", target="_top"]; "XIII.10" [label="XIII.10", fillcolor="#222244", tooltip="XIII.10: If an equilateral pentagon be inscribed in a circle, the square on the side of the pentagon is equal to the squares on the side of the hexagon and on that of the decagon inscribed…", URL="../10/index.html", target="_top"]; "XIII.11" [label="XIII.11", fillcolor="#222244", tooltip="XIII.11: If in a circle which has its diameter rational an equilateral pentagon be inscribed, the side of the pentagon is the irrational straight line called minor.", URL="../11/index.html", target="_top"]; "XIII.12" [label="XIII.12", fillcolor="#222244", tooltip="XIII.12: If an equilateral triangle be inscribed in a circle, the square on the side of the triangle is triple of the square on the radius of the circle.", URL="../12/index.html", target="_top"]; "XIII.13" [label="XIII.13", fillcolor="#222244", tooltip="XIII.13: To construct a pyramid, to comprehend it in a given sphere, and to prove that the square on the diameter of the sphere is one and a half times the square on the side of the…", URL="../13/index.html", target="_top"]; "XIII.14" [label="XIII.14", fillcolor="#222244", tooltip="XIII.14: To construct an octahedron and comprehend it in a sphere, as in the preceding case; and to prove that the square on the diameter of the sphere is double of the square on the side…", URL="../14/index.html", target="_top"]; "XIII.15" [label="XIII.15", fillcolor="#222244", tooltip="XIII.15: To construct a cube and comprehend it in a sphere, like the pyramid; and to prove that the square on the diameter of the sphere is triple of the square on the side of the cube.", URL="../15/index.html", target="_top"]; "XIII.16" [label="XIII.16", fillcolor="#222244", tooltip="XIII.16: To construct an icosahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the side of the icosahedron is the irrational straight line called minor.", URL="../16/index.html", target="_top"]; "XIII.17" [label="XIII.17", fillcolor="#222244", tooltip="XIII.17: To construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures, and to prove that the side of the dodecahedron is the irrational straight line called…", URL="../17/index.html", target="_top"]; "XIII.18" [label="XIII.18", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="XIII.18: To set out the sides of the five figures and to compare them with one another.", 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.12" -> "I.def.1"; "I.14" -> "I.def.1"; "I.23" -> "I.def.1"; "I.31" -> "I.def.1"; "II.5" -> "I.def.1"; "III.def.8" -> "I.def.1"; "III.10" -> "I.def.1"; "III.18" -> "I.def.1"; "III.19" -> "I.def.1"; "III.32" -> "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"; "XI.11" -> "I.def.1"; "XI.12" -> "I.def.1"; "XI.38" -> "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.def.3" -> "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"; "IV.def.7" -> "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.12" -> "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"; "I.46" -> "I.def.4"; "II.def.1" -> "I.def.4"; "II.2" -> "I.def.4"; "II.5" -> "I.def.4"; "II.7" -> "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.19" -> "I.def.4"; "III.24" -> "I.def.4"; "III.28" -> "I.def.4"; "III.29" -> "I.def.4"; "III.32" -> "I.def.4"; "IV.def.7" -> "I.def.4"; "IV.1" -> "I.def.4"; "VI.2" -> "I.def.4"; "VI.11" -> "I.def.4"; "VI.12" -> "I.def.4"; "VI.13" -> "I.def.4"; "VI.16" -> "I.def.4"; "VI.17" -> "I.def.4"; "VI.18" -> "I.def.4"; "VI.22" -> "I.def.4"; "VI.28" -> "I.def.4"; "VI.32" -> "I.def.4"; "X.def.2" -> "I.def.4"; "X.def.3" -> "I.def.4"; "X.def.4" -> "I.def.4"; "X.def.iii.1" -> "I.def.4"; "X.def.iii.4" -> "I.def.4"; "X.9" -> "I.def.4"; "X.16" -> "I.def.4"; "X.17" -> "I.def.4"; "X.18" -> "I.def.4"; "X.19" -> "I.def.4"; "X.20" -> "I.def.4"; "X.21" -> "I.def.4"; "X.22" -> "I.def.4"; "X.23" -> "I.def.4"; "X.30" -> "I.def.4"; "X.33" -> "I.def.4"; "X.73" -> "I.def.4"; "X.76" -> "I.def.4"; "X.94" -> "I.def.4"; "X.97" -> "I.def.4"; "XI.def.3" -> "I.def.4"; "XI.def.15" -> "I.def.4"; "XI.def.17" -> "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.11" -> "I.def.4"; "XI.12" -> "I.def.4"; "XIII.1" -> "I.def.4"; "XIII.3" -> "I.def.4"; "XIII.4" -> "I.def.4"; "XIII.5" -> "I.def.4"; "XIII.6" -> "I.def.4"; "XIII.8" -> "I.def.4"; "XIII.9" -> "I.def.4"; "XIII.11" -> "I.def.4"; "XIII.16" -> "I.def.4"; "XIII.17" -> "I.def.4"; "I.def.6" -> "I.def.5"; "I.def.7" -> "I.def.5"; "XI.def.17" -> "I.def.5"; "I.def.15" -> "I.def.6"; "I.def.8" -> "I.def.7"; "I.def.15" -> "I.def.7"; "I.def.23" -> "I.def.7"; "VIII.18" -> "I.def.7"; "XI.def.3" -> "I.def.7"; "XI.def.12" -> "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.11" -> "I.def.7"; "XI.12" -> "I.def.7"; "XI.38" -> "I.def.7"; "I.def.9" -> "I.def.8"; "I.def.10" -> "I.def.8"; "I.def.11" -> "I.def.8"; "I.def.12" -> "I.def.8"; "I.def.21" -> "I.def.8"; "I.def.22" -> "I.def.8"; "I.post.5" -> "I.def.8"; "I.4" -> "I.def.8"; "I.5" -> "I.def.8"; "I.6" -> "I.def.8"; "I.8" -> "I.def.8"; "I.13" -> "I.def.8"; "I.14" -> "I.def.8"; "I.15" -> "I.def.8"; "I.16" -> "I.def.8"; "I.17" -> "I.def.8"; "I.18" -> "I.def.8"; "I.19" -> "I.def.8"; "I.26" -> "I.def.8"; "I.27" -> "I.def.8"; "I.28" -> "I.def.8"; "I.29" -> "I.def.8"; "I.32" -> "I.def.8"; "I.34" -> "I.def.8"; "III.def.8" -> "I.def.8"; "III.def.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"; "III.32" -> "I.def.8"; "IV.def.2" -> "I.def.8"; "IV.def.3" -> "I.def.8"; "IV.def.6" -> "I.def.8"; "VI.def.1" -> "I.def.8"; "VI.4" -> "I.def.8"; "VI.6" -> "I.def.8"; "VI.14" -> "I.def.8"; "VI.15" -> "I.def.8"; "VI.26" -> "I.def.8"; "VI.32" -> "I.def.8"; "VI.33" -> "I.def.8"; "XIII.7" -> "I.def.8"; "XIII.8" -> "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.12" -> "I.def.10"; "I.13" -> "I.def.10"; "I.14" -> "I.def.10"; "I.17" -> "I.def.10"; "I.28" -> "I.def.10"; "I.29" -> "I.def.10"; "I.32" -> "I.def.10"; "I.47" -> "I.def.10"; "II.def.1" -> "I.def.10"; "III.1" -> "I.def.10"; "III.16" -> "I.def.10"; "III.18" -> "I.def.10"; "III.19" -> "I.def.10"; "III.22" -> "I.def.10"; "III.31" -> "I.def.10"; "VI.def.4" -> "I.def.10"; "VI.8" -> "I.def.10"; "XI.def.3" -> "I.def.10"; "XI.4" -> "I.def.10"; "XI.5" -> "I.def.10"; "XI.8" -> "I.def.10"; "XI.11" -> "I.def.10"; "XI.12" -> "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"; "I.6" -> "I.def.14"; "III.def.6" -> "I.def.14"; "III.def.10" -> "I.def.14"; "IV.def.2" -> "I.def.14"; "IV.def.3" -> "I.def.14"; "IV.def.4" -> "I.def.14"; "IV.def.5" -> "I.def.14"; "IV.def.6" -> "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"; "VI.21" -> "I.def.14"; "VI.25" -> "I.def.14"; "VI.28" -> "I.def.14"; "X.16" -> "I.def.14"; "X.17" -> "I.def.14"; "X.18" -> "I.def.14"; "XI.def.14" -> "I.def.14"; "XIII.3" -> "I.def.14"; "XIII.16" -> "I.def.14"; "XIII.17" -> "I.def.14"; "XIII.18" -> "I.def.14"; "I.def.16" -> "I.def.15"; "I.def.17" -> "I.def.15"; "I.def.18" -> "I.def.15"; "I.post.3" -> "I.def.15"; "I.1" -> "I.def.15"; "I.3" -> "I.def.15"; "III.def.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.19" -> "I.def.15"; "III.20" -> "I.def.15"; "III.21" -> "I.def.15"; "III.22" -> "I.def.15"; "III.31" -> "I.def.15"; "III.32" -> "I.def.15"; "IV.def.3" -> "I.def.15"; "IV.def.4" -> "I.def.15"; "IV.def.5" -> "I.def.15"; "IV.def.6" -> "I.def.15"; "IV.def.7" -> "I.def.15"; "IV.1" -> "I.def.15"; "IV.2" -> "I.def.15"; "IV.14" -> "I.def.15"; "IV.15" -> "I.def.15"; "XI.1" -> "I.def.15"; "XIII.9" -> "I.def.15"; "XIII.10" -> "I.def.15"; "XIII.11" -> "I.def.15"; "XIII.12" -> "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.19" -> "I.def.16"; "III.26" -> "I.def.16"; "III.27" -> "I.def.16"; "IV.14" -> "I.def.16"; "VI.33" -> "I.def.16"; "XI.def.17" -> "I.def.16"; "XI.1" -> "I.def.16"; "I.def.18" -> "I.def.17"; "I.34" -> "I.def.17"; "I.43" -> "I.def.17"; "II.def.2" -> "I.def.17"; "III.def.1" -> "I.def.17"; "III.16" -> "I.def.17"; "III.31" -> "I.def.17"; "IV.1" -> "I.def.17"; "VI.24" -> "I.def.17"; "VI.26" -> "I.def.17"; "XI.def.14" -> "I.def.17"; "XI.1" -> "I.def.17"; "XI.38" -> "I.def.17"; "XIII.11" -> "I.def.17"; "III.31" -> "I.def.18"; "XI.def.14" -> "I.def.18"; "XI.def.15" -> "I.def.18"; "XI.def.16" -> "I.def.18"; "I.def.20" -> "I.def.19"; "I.def.21" -> "I.def.19"; "I.def.22" -> "I.def.19"; "I.4" -> "I.def.19"; "I.6" -> "I.def.19"; "I.8" -> "I.def.19"; "I.16" -> "I.def.19"; "I.17" -> "I.def.19"; "I.18" -> "I.def.19"; "I.19" -> "I.def.19"; "I.20" -> "I.def.19"; "I.22" -> "I.def.19"; "I.26" -> "I.def.19"; "I.32" -> "I.def.19"; "I.37" -> "I.def.19"; "I.38" -> "I.def.19"; "I.39" -> "I.def.19"; "I.41" -> "I.def.19"; "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"; "IV.2" -> "I.def.19"; "VI.1" -> "I.def.19"; "VI.2" -> "I.def.19"; "VI.4" -> "I.def.19"; "VI.6" -> "I.def.19"; "VI.8" -> "I.def.19"; "VI.15" -> "I.def.19"; "VI.20" -> "I.def.19"; "VI.21" -> "I.def.19"; "VI.22" -> "I.def.19"; "VI.25" -> "I.def.19"; "VI.28" -> "I.def.19"; "VI.32" -> "I.def.19"; "X.def.4" -> "I.def.19"; "XI.2" -> "I.def.19"; "XIII.12" -> "I.def.19"; "I.def.22" -> "I.def.20"; "I.1" -> "I.def.20"; "I.5" -> "I.def.20"; "IV.14" -> "I.def.20"; "IV.15" -> "I.def.20"; "XI.def.26" -> "I.def.20"; "XI.def.27" -> "I.def.20"; "XI.def.28" -> "I.def.20"; "XIII.7" -> "I.def.20"; "XIII.8" -> "I.def.20"; "XIII.10" -> "I.def.20"; "XIII.11" -> "I.def.20"; "XIII.12" -> "I.def.20"; "I.47" -> "I.def.21"; "VI.8" -> "I.def.21"; "I.35" -> "I.def.22"; "I.46" -> "I.def.22"; "I.47" -> "I.def.22"; "II.2" -> "I.def.22"; "II.5" -> "I.def.22"; "II.7" -> "I.def.22"; "VI.17" -> "I.def.22"; "VIII.2" -> "I.def.22"; "VIII.11" -> "I.def.22"; "X.def.2" -> "I.def.22"; "X.def.4" -> "I.def.22"; "X.def.iii.1" -> "I.def.22"; "X.def.iii.4" -> "I.def.22"; "X.9" -> "I.def.22"; "X.16" -> "I.def.22"; "X.17" -> "I.def.22"; "X.18" -> "I.def.22"; "X.21" -> "I.def.22"; "X.22" -> "I.def.22"; "X.30" -> "I.def.22"; "X.33" -> "I.def.22"; "X.76" -> "I.def.22"; "X.97" -> "I.def.22"; "XI.def.25" -> "I.def.22"; "XIII.1" -> "I.def.22"; "XIII.3" -> "I.def.22"; "XIII.4" -> "I.def.22"; "XIII.10" -> "I.def.22"; "XIII.12" -> "I.def.22"; "XIII.13" -> "I.def.22"; "XIII.14" -> "I.def.22"; "XIII.15" -> "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"; "VI.32" -> "I.def.23"; "XI.6" -> "I.def.23"; "XI.7" -> "I.def.23"; "XI.8" -> "I.def.23"; "XI.9" -> "I.def.23"; "I.1" -> "I.post.1"; "I.2" -> "I.post.1"; "I.5" -> "I.post.1"; "I.7" -> "I.post.1"; "I.12" -> "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.12" -> "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.47" -> "I.cn.2"; "I.2" -> "I.cn.3"; "I.14" -> "I.cn.3"; "I.15" -> "I.cn.3"; "I.35" -> "I.cn.3"; "I.43" -> "I.cn.3"; "I.4" -> "I.cn.4"; "I.16" -> "I.cn.5"; "I.20" -> "I.cn.5"; "I.2" -> "I.1"; "I.10" -> "I.1"; "I.11" -> "I.1"; "I.3" -> "I.2"; "I.5" -> "I.3"; "I.9" -> "I.3"; "I.11" -> "I.3"; "I.16" -> "I.3"; "I.18" -> "I.3"; "I.22" -> "I.3"; "VI.11" -> "I.3"; "I.5" -> "I.4"; "I.10" -> "I.4"; "I.16" -> "I.4"; "I.26" -> "I.4"; "I.33" -> "I.4"; "I.34" -> "I.4"; "I.35" -> "I.4"; "I.47" -> "I.4"; "III.26" -> "I.4"; "III.29" -> "I.4"; "VI.6" -> "I.4"; "XI.4" -> "I.4"; "XI.6" -> "I.4"; "XI.38" -> "I.4"; "XIII.7" -> "I.4"; "XIII.8" -> "I.4"; "XIII.10" -> "I.4"; "XIII.13" -> "I.4"; "XIII.14" -> "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"; "IV.15" -> "I.5"; "XIII.7" -> "I.5"; "XIII.8" -> "I.5"; "XIII.9" -> "I.5"; "XIII.10" -> "I.5"; "IV.14" -> "I.6"; "XIII.7" -> "I.6"; "XIII.8" -> "I.6"; "I.8" -> "I.7"; "I.9" -> "I.8"; "I.11" -> "I.8"; "I.12" -> "I.8"; "I.23" -> "I.8"; "III.1" -> "I.8"; "III.28" -> "I.8"; "XI.4" -> "I.8"; "XI.6" -> "I.8"; "XIII.7" -> "I.8"; "XIII.17" -> "I.8"; "I.10" -> "I.9"; "I.12" -> "I.10"; "I.16" -> "I.10"; "I.13" -> "I.11"; "I.46" -> "I.11"; "XI.11" -> "I.11"; "XI.11" -> "I.12"; "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"; "I.47" -> "I.14"; "VI.14" -> "I.14"; "VI.15" -> "I.14"; "VI.32" -> "I.14"; "XI.38" -> "I.14"; "I.16" -> "I.15"; "I.28" -> "I.15"; "I.29" -> "I.15"; "I.44" -> "I.15"; "IV.15" -> "I.15"; "XI.4" -> "I.15"; "XI.38" -> "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"; "IV.2" -> "I.23"; "VI.6" -> "I.23"; "VI.18" -> "I.23"; "I.34" -> "I.26"; "XI.4" -> "I.26"; "XI.38" -> "I.26"; "XIII.10" -> "I.26"; "I.28" -> "I.27"; "I.31" -> "I.27"; "I.33" -> "I.27"; "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"; "I.46" -> "I.29"; "VI.32" -> "I.29"; "XI.8" -> "I.29"; "XI.38" -> "I.29"; "I.45" -> "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"; "I.46" -> "I.31"; "II.2" -> "I.31"; "II.5" -> "I.31"; "VI.11" -> "I.31"; "VI.12" -> "I.31"; "VI.26" -> "I.31"; "XI.11" -> "I.31"; "XI.12" -> "I.31"; "III.20" -> "I.32"; "III.22" -> "I.32"; "III.31" -> "I.32"; "III.32" -> "I.32"; "IV.2" -> "I.32"; "IV.15" -> "I.32"; "VI.6" -> "I.32"; "VI.8" -> "I.32"; "VI.18" -> "I.32"; "VI.20" -> "I.32"; "VI.32" -> "I.32"; "XIII.8" -> "I.32"; "XIII.9" -> "I.32"; "XIII.10" -> "I.32"; "XIII.11" -> "I.32"; "I.36" -> "I.33"; "I.45" -> "I.33"; "XI.38" -> "I.33"; "XIII.16" -> "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"; "I.46" -> "I.34"; "VI.4" -> "I.34"; "I.36" -> "I.35"; "I.37" -> "I.35"; "I.38" -> "I.36"; "II.5" -> "I.36"; "VI.28" -> "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"; "I.47" -> "I.41"; "VI.1" -> "I.41"; "I.44" -> "I.42"; "I.45" -> "I.42"; "I.44" -> "I.43"; "II.5" -> "I.43"; "II.7" -> "I.43"; "I.45" -> "I.44"; "VI.25" -> "I.44"; "VI.25" -> "I.45"; "I.47" -> "I.46"; "II.2" -> "I.46"; "II.5" -> "I.46"; "II.7" -> "I.46"; "X.13" -> "I.47"; "X.30" -> "I.47"; "X.33" -> "I.47"; "XIII.12" -> "I.47"; "XIII.14" -> "I.47"; "XIII.15" -> "I.47"; "XIII.17" -> "I.47"; "II.2" -> "II.def.1"; "II.5" -> "II.def.1"; "II.7" -> "II.def.1"; "VI.16" -> "II.def.1"; "VI.17" -> "II.def.1"; "X.16" -> "II.def.1"; "X.19" -> "II.def.1"; "X.21" -> "II.def.1"; "X.33" -> "II.def.1"; "X.76" -> "II.def.1"; "XIII.3" -> "II.def.1"; "II.5" -> "II.def.2"; "II.7" -> "II.def.2"; "VI.28" -> "II.def.2"; "XIII.1" -> "II.def.2"; "XIII.3" -> "II.def.2"; "XIII.4" -> "II.def.2"; "XIII.10" -> "II.2"; "X.17" -> "II.5"; "X.73" -> "II.7"; "X.76" -> "II.7"; "X.97" -> "II.7"; "III.26" -> "III.def.1"; "III.27" -> "III.def.1"; "III.28" -> "III.def.1"; "III.29" -> "III.def.1"; "VI.33" -> "III.def.1"; "III.16" -> "III.def.2"; "III.18" -> "III.def.2"; "III.19" -> "III.def.2"; "III.32" -> "III.def.2"; "III.32" -> "III.def.6"; "III.16" -> "III.def.7"; "III.31" -> "III.def.8"; "III.26" -> "III.def.9"; "III.27" -> "III.def.9"; "VI.33" -> "III.def.9"; "III.20" -> "III.def.10"; "III.24" -> "III.def.11"; "III.26" -> "III.def.11"; "III.10" -> "III.1"; "XIII.13" -> "III.1"; "III.10" -> "III.5"; "III.24" -> "III.10"; "IV.2" -> "III.16"; "III.19" -> "III.18"; "III.32" -> "III.19"; "III.21" -> "III.20"; "III.27" -> "III.20"; "III.22" -> "III.21"; "III.31" -> "III.22"; "III.32" -> "III.22"; "III.26" -> "III.24"; "III.27" -> "III.26"; "III.28" -> "III.26"; "IV.15" -> "III.26"; "XIII.10" -> "III.26"; "III.29" -> "III.27"; "IV.15" -> "III.27"; "VI.33" -> "III.27"; "XIII.8" -> "III.28"; "IV.15" -> "III.29"; "III.32" -> "III.31"; "VI.13" -> "III.31"; "X.13" -> "III.31"; "X.30" -> "III.31"; "XIII.12" -> "III.31"; "XIII.13" -> "III.31"; "XIII.14" -> "III.31"; "XIII.16" -> "III.31"; "IV.2" -> "III.32"; "XIII.8" -> "IV.def.2"; "IV.2" -> "IV.def.3"; "IV.15" -> "IV.def.3"; "XIII.10" -> "IV.def.3"; "XIII.12" -> "IV.def.3"; "IV.15" -> "IV.def.4"; "IV.15" -> "IV.def.5"; "IV.14" -> "IV.def.6"; "IV.1" -> "IV.def.7"; "X.13" -> "IV.1"; "XIII.13" -> "IV.2"; "XIII.8" -> "IV.14"; "XIII.9" -> "IV.15"; "XIII.10" -> "IV.15"; "XIII.12" -> "IV.15"; "XIII.16" -> "IV.15"; "XIII.18" -> "IV.15"; "V.def.2" -> "V.def.1"; "VII.def.3" -> "V.def.1"; "VII.def.4" -> "V.def.1"; "VII.def.5" -> "V.def.1"; "VII.def.11" -> "V.def.1"; "VII.def.12" -> "V.def.1"; "VII.1" -> "V.def.1"; "VII.2" -> "V.def.1"; "VII.3" -> "V.def.1"; "VII.15" -> "V.def.1"; "VII.20" -> "V.def.1"; "VII.23" -> "V.def.1"; "VII.34" -> "V.def.1"; "VII.35" -> "V.def.1"; "X.def.1" -> "V.def.1"; "X.def.2" -> "V.def.1"; "V.def.5" -> "V.def.2"; "V.def.7" -> "V.def.2"; "V.1" -> "V.def.2"; "V.2" -> "V.def.2"; "V.3" -> "V.def.2"; "V.4" -> "V.def.2"; "V.15" -> "V.def.2"; "VII.5" -> "V.def.2"; "V.def.17" -> "V.def.3"; "V.10" -> "V.def.3"; "V.11" -> "V.def.3"; "VI.20" -> "V.def.3"; "VII.33" -> "V.def.3"; "VIII.2" -> "V.def.3"; "VIII.4" -> "V.def.3"; "VIII.11" -> "V.def.3"; "VIII.18" -> "V.def.3"; "VIII.26" -> "V.def.3"; "X.5" -> "V.def.3"; "X.6" -> "V.def.3"; "X.7" -> "V.def.3"; "X.8" -> "V.def.3"; "X.9" -> "V.def.3"; "V.8" -> "V.def.4"; "V.10" -> "V.def.4"; "V.def.6" -> "V.def.5"; "V.4" -> "V.def.5"; "V.7" -> "V.def.5"; "V.9" -> "V.def.5"; "V.11" -> "V.def.5"; "V.12" -> "V.def.5"; "V.13" -> "V.def.5"; "V.14" -> "V.def.5"; "V.15" -> "V.def.5"; "V.16" -> "V.def.5"; "V.20" -> "V.def.5"; "V.22" -> "V.def.5"; "VI.1" -> "V.def.5"; "VI.20" -> "V.def.5"; "VI.33" -> "V.def.5"; "VII.14" -> "V.def.5"; "VII.17" -> "V.def.5"; "VII.18" -> "V.def.5"; "VII.20" -> "V.def.5"; "VII.21" -> "V.def.5"; "VII.22" -> "V.def.5"; "VII.33" -> "V.def.5"; "VIII.1" -> "V.def.5"; "V.def.9" -> "V.def.6"; "V.7" -> "V.def.6"; "V.12" -> "V.def.6"; "V.16" -> "V.def.6"; "V.17" -> "V.def.6"; "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.13" -> "V.def.6"; "VI.16" -> "V.def.6"; "VI.17" -> "V.def.6"; "VI.22" -> "V.def.6"; "VI.32" -> "V.def.6"; "VII.def.21" -> "V.def.6"; "VII.13" -> "V.def.6"; "VII.19" -> "V.def.6"; "VIII.1" -> "V.def.6"; "VIII.2" -> "V.def.6"; "VIII.4" -> "V.def.6"; "VIII.11" -> "V.def.6"; "VIII.18" -> "V.def.6"; "X.11" -> "V.def.6"; "V.8" -> "V.def.7"; "V.10" -> "V.def.7"; "V.13" -> "V.def.7"; "VI.19" -> "V.def.9"; "VIII.11" -> "V.def.9"; "VIII.18" -> "V.def.9"; "XIII.16" -> "V.def.9"; "XIII.18" -> "V.def.9"; "V.def.12" -> "V.def.11"; "V.def.13" -> "V.def.11"; "V.def.14" -> "V.def.11"; "V.def.15" -> "V.def.11"; "V.def.16" -> "V.def.11"; "V.12" -> "V.def.11"; "VI.19" -> "V.def.11"; "VII.12" -> "V.def.11"; "V.16" -> "V.def.12"; "VII.9" -> "V.def.12"; "VII.10" -> "V.def.12"; "VII.13" -> "V.def.12"; "VII.15" -> "V.def.12"; "V.7" -> "V.def.13"; "V.17" -> "V.def.14"; "V.18" -> "V.def.14"; "V.17" -> "V.def.15"; "V.18" -> "V.def.15"; "X.17" -> "V.def.15"; "XIII.12" -> "V.def.15"; "V.19" -> "V.def.16"; "XIII.13" -> "V.def.16"; "V.3" -> "V.def.17"; "V.20" -> "V.def.17"; "V.22" -> "V.def.17"; "VII.14" -> "V.def.17"; "V.8" -> "V.1"; "V.12" -> "V.1"; "V.17" -> "V.1"; "V.3" -> "V.2"; "V.17" -> "V.2"; "V.4" -> "V.3"; "V.22" -> "V.4"; "V.10" -> "V.7"; "V.15" -> "V.7"; "VI.2" -> "V.7"; "VI.14" -> "V.7"; "VI.15" -> "V.7"; "VII.19" -> "V.7"; "X.5" -> "V.7"; "X.6" -> "V.7"; "V.9" -> "V.8"; "V.10" -> "V.8"; "V.14" -> "V.8"; "V.20" -> "V.8"; "VI.2" -> "V.9"; "VI.6" -> "V.9"; "VI.14" -> "V.9"; "VI.15" -> "V.9"; "VI.22" -> "V.9"; "VI.26" -> "V.9"; "VII.19" -> "V.9"; "X.6" -> "V.9"; "V.14" -> "V.10"; "V.20" -> "V.10"; "V.16" -> "V.11"; "V.18" -> "V.11"; "V.19" -> "V.11"; "VI.1" -> "V.11"; "VI.2" -> "V.11"; "VI.6" -> "V.11"; "VI.14" -> "V.11"; "VI.15" -> "V.11"; "VI.19" -> "V.11"; "VI.22" -> "V.11"; "VI.26" -> "V.11"; "X.6" -> "V.11"; "X.12" -> "V.11"; "X.94" -> "V.11"; "V.15" -> "V.12"; "VI.20" -> "V.12"; "V.14" -> "V.13"; "V.20" -> "V.13"; "V.16" -> "V.14"; "V.18" -> "V.14"; "XIII.5" -> "V.14"; "XIII.8" -> "V.14"; "V.16" -> "V.15"; "VI.1" -> "V.15"; "XIII.17" -> "V.15"; "V.19" -> "V.16"; "VI.4" -> "V.16"; "VI.19" -> "V.16"; "VI.24" -> "V.16"; "VI.25" -> "V.16"; "V.18" -> "V.17"; "V.19" -> "V.17"; "VI.24" -> "V.18"; "XIII.11" -> "V.18"; "X.30" -> "V.19"; "XIII.11" -> "V.19"; "V.22" -> "V.20"; "VI.4" -> "V.22"; "VI.20" -> "V.22"; "VI.22" -> "V.22"; "VI.24" -> "V.22"; "X.5" -> "V.22"; "X.6" -> "V.22"; "X.12" -> "V.22"; "VI.8" -> "VI.def.1"; "VI.18" -> "VI.def.1"; "VI.19" -> "VI.def.1"; "VI.20" -> "VI.def.1"; "VI.21" -> "VI.def.1"; "VI.22" -> "VI.def.1"; "VI.24" -> "VI.def.1"; "VI.14" -> "VI.def.2"; "VI.15" -> "VI.def.2"; "XIII.1" -> "VI.def.3"; "XIII.3" -> "VI.def.3"; "XIII.4" -> "VI.def.3"; "XIII.5" -> "VI.def.3"; "XIII.6" -> "VI.def.3"; "XIII.8" -> "VI.def.3"; "XIII.9" -> "VI.def.3"; "VI.1" -> "VI.def.4"; "VI.2" -> "VI.1"; "VI.14" -> "VI.1"; "VI.15" -> "VI.1"; "VI.19" -> "VI.1"; "VI.20" -> "VI.1"; "VI.25" -> "VI.1"; "X.19" -> "VI.1"; "X.20" -> "VI.1"; "X.21" -> "VI.1"; "X.23" -> "VI.1"; "X.94" -> "VI.1"; "X.97" -> "VI.1"; "XIII.1" -> "VI.1"; "VI.4" -> "VI.2"; "VI.11" -> "VI.2"; "VI.12" -> "VI.2"; "VI.24" -> "VI.2"; "VI.6" -> "VI.4"; "VI.8" -> "VI.4"; "VI.18" -> "VI.4"; "VI.20" -> "VI.4"; "XIII.8" -> "VI.4"; "XIII.9" -> "VI.4"; "XIII.10" -> "VI.4"; "XIII.18" -> "VI.4"; "VI.20" -> "VI.6"; "VI.32" -> "VI.6"; "VI.13" -> "VI.8"; "XIII.13" -> "VI.8"; "XIII.16" -> "VI.8"; "XIII.18" -> "VI.8"; "VI.19" -> "VI.11"; "VI.22" -> "VI.11"; "VI.22" -> "VI.12"; "VI.25" -> "VI.13"; "VI.16" -> "VI.14"; "X.22" -> "VI.14"; "VI.19" -> "VI.15"; "VI.17" -> "VI.16"; "X.94" -> "VI.17"; "X.97" -> "VI.17"; "XIII.1" -> "VI.17"; "XIII.4" -> "VI.17"; "XIII.5" -> "VI.17"; "XIII.6" -> "VI.17"; "XIII.10" -> "VI.17"; "XIII.13" -> "VI.17"; "VI.22" -> "VI.18"; "VI.25" -> "VI.18"; "VI.28" -> "VI.18"; "VI.20" -> "VI.19"; "VI.22" -> "VI.19"; "VI.25" -> "VI.19"; "X.6" -> "VI.19"; "X.9" -> "VI.20"; "XIII.18" -> "VI.20"; "VI.24" -> "VI.21"; "VI.28" -> "VI.21"; "X.22" -> "VI.22"; "VI.26" -> "VI.24"; "VI.28" -> "VI.25"; "VI.28" -> "VI.26"; "X.94" -> "VI.26"; "X.33" -> "VI.28"; "XIII.17" -> "VI.32"; "XIII.8" -> "VI.33"; "XIII.9" -> "VI.33"; "XIII.10" -> "VI.33"; "VII.def.2" -> "VII.def.1"; "VII.def.11" -> "VII.def.1"; "VII.def.12" -> "VII.def.1"; "VII.def.15" -> "VII.def.1"; "VII.1" -> "VII.def.1"; "VII.15" -> "VII.def.1"; "VII.def.3" -> "VII.def.2"; "VII.def.5" -> "VII.def.2"; "VII.def.15" -> "VII.def.2"; "VII.def.16" -> "VII.def.2"; "VII.def.18" -> "VII.def.2"; "VII.1" -> "VII.def.2"; "VII.2" -> "VII.def.2"; "VII.3" -> "VII.def.2"; "VII.4" -> "VII.def.2"; "VII.5" -> "VII.def.2"; "VII.6" -> "VII.def.2"; "VII.9" -> "VII.def.2"; "VII.10" -> "VII.def.2"; "VII.12" -> "VII.def.2"; "VII.13" -> "VII.def.2"; "VII.14" -> "VII.def.2"; "VII.15" -> "VII.def.2"; "VII.16" -> "VII.def.2"; "VII.17" -> "VII.def.2"; "VII.18" -> "VII.def.2"; "VII.19" -> "VII.def.2"; "VII.20" -> "VII.def.2"; "VII.21" -> "VII.def.2"; "VII.22" -> "VII.def.2"; "VII.23" -> "VII.def.2"; "VII.24" -> "VII.def.2"; "VII.25" -> "VII.def.2"; "VII.26" -> "VII.def.2"; "VII.27" -> "VII.def.2"; "VII.33" -> "VII.def.2"; "VII.34" -> "VII.def.2"; "VII.35" -> "VII.def.2"; "VIII.1" -> "VII.def.2"; "VIII.2" -> "VII.def.2"; "VIII.4" -> "VII.def.2"; "VIII.11" -> "VII.def.2"; "VIII.18" -> "VII.def.2"; "X.5" -> "VII.def.2"; "X.6" -> "VII.def.2"; "X.7" -> "VII.def.2"; "X.8" -> "VII.def.2"; "VII.def.20" -> "VII.def.3"; "VII.4" -> "VII.def.3"; "VII.5" -> "VII.def.3"; "VII.9" -> "VII.def.3"; "VII.10" -> "VII.def.3"; "X.17" -> "VII.def.3"; "X.18" -> "VII.def.3"; "XI.1" -> "VII.def.3"; "XIII.12" -> "VII.def.3"; "VII.def.20" -> "VII.def.4"; "VII.4" -> "VII.def.4"; "VII.6" -> "VII.def.4"; "VII.9" -> "VII.def.4"; "VII.10" -> "VII.def.4"; "VII.def.20" -> "VII.def.5"; "VII.5" -> "VII.def.5"; "VII.34" -> "VII.def.11"; "VIII.1" -> "VII.def.11"; "VII.1" -> "VII.def.12"; "VII.2" -> "VII.def.12"; "VII.3" -> "VII.def.12"; "VII.21" -> "VII.def.12"; "VII.22" -> "VII.def.12"; "VII.23" -> "VII.def.12"; "VII.24" -> "VII.def.12"; "VII.25" -> "VII.def.12"; "VII.26" -> "VII.def.12"; "VII.27" -> "VII.def.12"; "VIII.1" -> "VII.def.12"; "VII.def.16" -> "VII.def.15"; "VII.def.18" -> "VII.def.15"; "VII.16" -> "VII.def.15"; "VII.17" -> "VII.def.15"; "VII.18" -> "VII.def.15"; "VII.22" -> "VII.def.15"; "VII.24" -> "VII.def.15"; "VII.27" -> "VII.def.15"; "VII.33" -> "VII.def.15"; "VII.34" -> "VII.def.15"; "VIII.18" -> "VII.def.16"; "VIII.11" -> "VII.def.18"; "VIII.26" -> "VII.def.18"; "X.9" -> "VII.def.18"; "VII.12" -> "VII.def.20"; "VII.13" -> "VII.def.20"; "VII.17" -> "VII.def.20"; "VII.20" -> "VII.def.20"; "VII.33" -> "VII.def.20"; "VIII.4" -> "VII.def.20"; "X.5" -> "VII.def.20"; "X.6" -> "VII.def.20"; "VIII.18" -> "VII.def.21"; "VIII.26" -> "VII.def.21"; "VII.2" -> "VII.1"; "VII.3" -> "VII.2"; "VII.4" -> "VII.2"; "VII.33" -> "VII.3"; "VII.20" -> "VII.4"; "VII.6" -> "VII.5"; "VII.9" -> "VII.5"; "VII.10" -> "VII.5"; "VII.12" -> "VII.5"; "VII.9" -> "VII.6"; "VII.10" -> "VII.6"; "VII.12" -> "VII.6"; "VII.10" -> "VII.9"; "VII.13" -> "VII.10"; "VII.15" -> "VII.12"; "VII.20" -> "VII.12"; "VII.14" -> "VII.13"; "VII.17" -> "VII.13"; "VII.20" -> "VII.13"; "VIII.4" -> "VII.13"; "VIII.18" -> "VII.13"; "VIII.1" -> "VII.14"; "VII.16" -> "VII.15"; "VII.18" -> "VII.16"; "VII.21" -> "VII.16"; "VII.24" -> "VII.16"; "VII.33" -> "VII.16"; "VII.34" -> "VII.16"; "VII.18" -> "VII.17"; "VII.19" -> "VII.17"; "VII.22" -> "VII.17"; "VII.34" -> "VII.17"; "VIII.2" -> "VII.17"; "VIII.11" -> "VII.17"; "VIII.18" -> "VII.17"; "VII.19" -> "VII.18"; "VIII.2" -> "VII.18"; "VIII.11" -> "VII.18"; "VII.24" -> "VII.19"; "VII.33" -> "VII.19"; "VII.34" -> "VII.19"; "VII.21" -> "VII.20"; "VII.24" -> "VII.20"; "VII.34" -> "VII.20"; "VIII.1" -> "VII.20"; "VIII.4" -> "VII.20"; "VII.24" -> "VII.21"; "VII.33" -> "VII.21"; "VII.34" -> "VII.21"; "VIII.1" -> "VII.21"; "VIII.2" -> "VII.22"; "VII.24" -> "VII.23"; "VII.25" -> "VII.24"; "VII.26" -> "VII.24"; "VII.27" -> "VII.25"; "VII.27" -> "VII.26"; "VIII.2" -> "VII.27"; "VII.34" -> "VII.33"; "VIII.26" -> "VII.33"; "VIII.4" -> "VII.34"; "VIII.4" -> "VII.35"; "VIII.4" -> "VIII.def.20"; "VIII.2" -> "VIII.1"; "VIII.26" -> "VIII.2"; "X.12" -> "VIII.4"; "X.9" -> "VIII.11"; "VIII.26" -> "VIII.18"; "X.9" -> "VIII.26"; "X.def.3" -> "X.def.1"; "X.def.4" -> "X.def.1"; "X.def.iii.1" -> "X.def.1"; "X.def.iii.4" -> "X.def.1"; "X.5" -> "X.def.1"; "X.6" -> "X.def.1"; "X.7" -> "X.def.1"; "X.8" -> "X.def.1"; "X.9" -> "X.def.1"; "X.11" -> "X.def.1"; "X.12" -> "X.def.1"; "X.13" -> "X.def.1"; "X.15" -> "X.def.1"; "X.16" -> "X.def.1"; "X.17" -> "X.def.1"; "X.18" -> "X.def.1"; "X.19" -> "X.def.1"; "X.20" -> "X.def.1"; "X.22" -> "X.def.1"; "X.23" -> "X.def.1"; "X.30" -> "X.def.1"; "X.73" -> "X.def.1"; "X.9" -> "X.def.2"; "X.21" -> "X.def.2"; "X.30" -> "X.def.2"; "X.33" -> "X.def.2"; "X.76" -> "X.def.2"; "X.def.iii.1" -> "X.def.3"; "X.def.iii.4" -> "X.def.3"; "X.19" -> "X.def.3"; "X.20" -> "X.def.3"; "X.21" -> "X.def.3"; "X.22" -> "X.def.3"; "X.23" -> "X.def.3"; "X.30" -> "X.def.3"; "X.33" -> "X.def.3"; "X.73" -> "X.def.3"; "X.76" -> "X.def.3"; "X.94" -> "X.def.3"; "X.97" -> "X.def.3"; "XIII.6" -> "X.def.3"; "XIII.11" -> "X.def.3"; "XIII.16" -> "X.def.3"; "XIII.17" -> "X.def.3"; "X.19" -> "X.def.4"; "X.20" -> "X.def.4"; "X.21" -> "X.def.4"; "X.22" -> "X.def.4"; "X.73" -> "X.def.4"; "XIII.6" -> "X.def.4"; "X.97" -> "X.def.iii.1"; "X.94" -> "X.def.iii.4"; "XIII.11" -> "X.def.iii.4"; "X.8" -> "X.5"; "X.9" -> "X.5"; "X.11" -> "X.5"; "X.12" -> "X.5"; "X.7" -> "X.6"; "X.9" -> "X.6"; "X.11" -> "X.6"; "X.12" -> "X.6"; "X.17" -> "X.6"; "X.18" -> "X.6"; "X.30" -> "X.6"; "X.73" -> "X.6"; "XIII.6" -> "X.6"; "X.11" -> "X.7"; "X.11" -> "X.8"; "X.30" -> "X.9"; "XIII.6" -> "X.9"; "XIII.11" -> "X.9"; "X.19" -> "X.11"; "X.20" -> "X.11"; "X.21" -> "X.11"; "X.22" -> "X.11"; "X.23" -> "X.11"; "X.73" -> "X.11"; "X.94" -> "X.11"; "X.97" -> "X.11"; "X.13" -> "X.12"; "X.17" -> "X.12"; "XIII.11" -> "X.12"; "X.18" -> "X.13"; "X.22" -> "X.13"; "X.23" -> "X.13"; "X.73" -> "X.13"; "X.17" -> "X.15"; "X.73" -> "X.15"; "XIII.11" -> "X.15"; "X.18" -> "X.16"; "X.73" -> "X.16"; "X.76" -> "X.16"; "X.18" -> "X.17"; "X.97" -> "X.17"; "X.33" -> "X.18"; "X.94" -> "X.18"; "X.94" -> "X.19"; "X.97" -> "X.20"; "X.22" -> "X.21"; "X.23" -> "X.21"; "X.33" -> "X.21"; "X.94" -> "X.21"; "X.23" -> "X.22"; "X.97" -> "X.22"; "X.33" -> "X.23"; "X.33" -> "X.30"; "X.76" -> "X.33"; "X.97" -> "X.73"; "XIII.6" -> "X.73"; "XIII.11" -> "X.73"; "X.94" -> "X.76"; "XIII.11" -> "X.94"; "XIII.6" -> "X.97"; "XI.def.12" -> "XI.def.1"; "XI.def.25" -> "XI.def.1"; "XI.def.26" -> "XI.def.1"; "XI.def.27" -> "XI.def.1"; "XI.def.28" -> "XI.def.1"; "XI.4" -> "XI.def.3"; "XI.5" -> "XI.def.3"; "XI.6" -> "XI.def.3"; "XI.8" -> "XI.def.3"; "XI.11" -> "XI.def.3"; "XIII.13" -> "XI.def.3"; "XIII.15" -> "XI.def.3"; "XIII.13" -> "XI.def.12"; "XIII.15" -> "XI.def.12"; "XI.def.17" -> "XI.def.14"; "XIII.13" -> "XI.def.14"; "XIII.14" -> "XI.def.14"; "XIII.15" -> "XI.def.14"; "XIII.16" -> "XI.def.14"; "XIII.17" -> "XI.def.14"; "XIII.13" -> "XI.def.15"; "XIII.14" -> "XI.def.15"; "XIII.15" -> "XI.def.15"; "XIII.16" -> "XI.def.15"; "XIII.17" -> "XI.def.16"; "XIII.13" -> "XI.def.17"; "XIII.14" -> "XI.def.17"; "XIII.15" -> "XI.def.17"; "XIII.16" -> "XI.def.17"; "XI.38" -> "XI.def.25"; "XIII.15" -> "XI.def.25"; "XIII.14" -> "XI.def.26"; "XIII.16" -> "XI.def.27"; "XIII.17" -> "XI.def.28"; "XI.2" -> "XI.1"; "XIII.17" -> "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.11" -> "XI.4"; "XI.6" -> "XI.5"; "XI.9" -> "XI.6"; "XIII.16" -> "XI.6"; "XIII.17" -> "XI.6"; "XI.8" -> "XI.7"; "XI.9" -> "XI.8"; "XI.11" -> "XI.8"; "XI.12" -> "XI.8"; "XI.38" -> "XI.9"; "XI.12" -> "XI.11"; "XIII.13" -> "XI.12"; "XIII.14" -> "XI.12"; "XIII.17" -> "XI.38"; "XIII.6" -> "XIII.1"; "XIII.11" -> "XIII.1"; "XIII.16" -> "XIII.3"; "XIII.17" -> "XIII.4"; "XIII.17" -> "XIII.5"; "XIII.17" -> "XIII.6"; "XIII.17" -> "XIII.7"; "XIII.11" -> "XIII.8"; "XIII.16" -> "XIII.9"; "XIII.18" -> "XIII.9"; "XIII.16" -> "XIII.10"; "XIII.18" -> "XIII.10"; "XIII.16" -> "XIII.11"; "XIII.13" -> "XIII.12"; "XIII.18" -> "XIII.13"; "XIII.18" -> "XIII.14"; "XIII.17" -> "XIII.15"; "XIII.18" -> "XIII.15"; "XIII.18" -> "XIII.16"; "XIII.18" -> "XIII.17"; }

See Commentary on Proposition XIII.18.