PROPOSITION 17.

Given two spheres about the same centre, to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface.

Let two spheres be conceived about the same centre A ; thus it is required to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface.

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

Let the spheres be cut by any plane through the centre ; then the sections will be circles, inasmuch as the sphere was produced by the diameter remaining fixed and the semicircle being carried round it ; [XI. Def. 14] hence, in whatever position we conceive the semicircle to be, the plane carried through it will produce a circle on the circumference of the sphere.

And it is manifest that this circle is the greatest possible, inasmuch as the diameter of the sphere, which is of course the diameter both of the semicircle and of the circle, is greater than all the straight lines drawn across in the circle or the sphere.

Let then BCDE be the circle in the greater sphere, and FGH the circle in the lesser sphere ; let two diameters in them, BD, CE, be drawn at right angles to one another ; then, given the two circles BCDE, FGH about the same centre, let there be inscribed in the greater circle BCDE an equilateral polygon with an even number of sides which does not touch the lesser circle FGH, let BK, KL, LM, ME be its sides in the quadrant BE, let KA be joined and carried through to N, let AO be set up from the point A at right angles to the plane of the circle BCDE, and let it meet the surface of the sphere at O, and through AO and each of the straight lines BD, KN let planes be carried ; they will then make greatest circles on the surface of the sphere, for the reason stated. Let them make such, and in them let BOD, KON be the semicircles on BD, KN.

Now, since OA is at right angles to the plane of the circle BCDE,
therefore all the planes through OA are also at right angles to the plane of the circle BCDE ; [XI. 18] hence the semicircles BOD, KON are also at right angles to the plane of the circle BCDE.
And, since the semicircles BED, BOD, KON are equal, for they are on the equal diameters BD, KN,
therefore the quadrants BE, BO, KO are also equal to one another.
Therefore there are as many straight lines in the quadrants BO, KO equal to the straight lines BK, KL, LM, ME as there are sides of the polygon in the quadrant BE.

Let them be inscribed, and let them be BP, PQ, QR, RO and KS, ST, TU, UO, let SP, TQ, UR be joined, and from P, S let perpendiculars be drawn to the plane of the circle BCDE ; [XI. 11] these will fall on BD, KN, the common sections of the planes, inasmuch as the planes of BOD, KON are also at right angles to the plane of the circle BCDE. [cf. XI. Def. 4] Let them so fall, and let them be PV, SW, and let WV be joined.

Now since, in the equal semicircles BOD, KON, equal straight lines BP, KS have been cut off, and the perpendiculars PV, SW have been drawn,
[III. 27, I. 26]therefore PV is equal to SW, and BV to KW.
But the whole BA is also equal to the whole KA ;
therefore the remainder VA is also equal to the remainder WA ; therefore, as BV is to VA, so is KW to WA ; therefore WV is parallel to KB. [VI. 2]
And, since each of the straight lines PV, SW is at right angles to the plane of the circle BCDE,
[XI. 6]therefore PV is parallel to SW.
But it was also proved equal to it ;
[I. 33]therefore WV, SP are also equal and parallel.
And, since WV is parallel to SP, while WV is parallel to KB,
[XI. 9]therefore SP is also parallel to KB.
And BP, KS join their extremities ;
therefore the quadrilateral KBPS is in one plane, inasmuch as, 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 parallels. [XI. 7]
For the same reason each of the quadrilaterals SPQT, TQRU is also in one plane.
[XI. 2]But the triangle URO is also in one plane.

If then we conceive straight lines joined from the points P, S, Q, T, R, U to A, there will be constructed a certain polyhedral solid figure between the circumferences BO, KO, consisting of pyramids of which the quadrilaterals KBPS, SPQT, TQRU and the triangle URO are the bases and the point A the vertex.

And, if we make the same construction in the case of each of the sides KL, LM, ME as in the case of BK, and further in the case of the remaining three quadrants, there will be constructed a certain polyhedral figure inscribed in the sphere and contained by pyramids, of which the said quadrilaterals and the triangle URO, and the others corresponding to them, are the bases and the point A the vertex.

I say that the said polyhedron will not touch the lesser sphere at the surface on which the circle FGH is.

Let AX be drawn from the point A perpendicular to the plane of the quadrilateral KBPS, and let it meet the plane at the point X ; [XI. 11] let XB, XK be joined.

Then, since AX is at right angles to the plane of the quadrilateral KBPS,
therefore it is also at right angles to all the straight lines which meet it and are in the plane of the quadrilateral. [XI. Def. 3] Therefore AX is at right angles to each of the straight lines BX, XK.
And, since AB is equal to AK,
the square on AB is also equal to the square on AK.
And the squares on AX, XB are equal to the square on AB, for the angle at X is right ; [I. 47] and the squares on AX, XK are equal to the square on AK. [*id.*]
Therefore the squares on AX, XB are equal to the squares on AX, XK.

Let the square on AX be subtracted from each ;

therefore the remainder, the square on BX, is equal to the remainder, the square on XK ; therefore BX is equal to XK.
Similarly we can prove that the straight lines joined from X to P, S are equal to each of the straight lines BX, XK.
Therefore the circle described with centre X and distance one of the straight lines XB, XK will pass through P, S also, and KBPS will be a quadrilateral in a circle.
Now, since KB is greater than WV, while WV is equal to SP,
therefore KB is greater than SP.
But KB is equal to each of the straight lines KS, BP;
therefore each of the straight lines KS, BP is greater than SP.
And, since KBPS is a quadrilateral in a circle, and KB, BP, KS are equal, and PS less, and BX is the radius of the circle,
therefore the square on KB is greater than double of the square on BX.

Let KZ be drawn from K perpendicular to BV.

Then, since BD is less than double of DZ, and, as BD is to DZ, so is the rectangle DB, BZ to the rectangle DZ, ZB, if a square be described upon BZ and the parallelogram on ZD be completed,
then the rectangle DB, BZ is also less than double of the rectangle DZ, ZB.
And, if KD be joined, the rectangle DB, BZ is equal to the square on BK, and the rectangle DZ, ZB equal to the square on KZ; [III. 31, VI. 8 and Por.]
therefore the square on KB is less than double of the square on KZ.
But the square on KB is greater than double of the square on BX;
therefore the square on KZ is greater than the square on BX.
And, since BA is equal to KA,
the square on BA is equal to the square on AK.
And the squares on BX, XA are equal to the square on BA, and the squares on KZ, ZA equal to the square on KA; [I. 47]
therefore the squares on BX, XA are equal to the squares on KZ, ZA,
and of these the square on KZ is greater than the square on BX; therefore the remainder, the square on ZA, is less than the square on XA.
Therefore AX is greater than AZ; therefore AX is much greater than AG.
And AX is the perpendicular on one base of the polyhedron, and AG on the surface of the lesser sphere;
hence the polyhedron will not touch the lesser sphere on its surface.
Therefore, given two spheres about the same centre, a polyhedral solid has been inscribed in the greater sphere which does not touch the lesser sphere at its surface. Q. E. F.
PORISM. But if in another sphere also a polyhedral solid be inscribed similar to the solid in the sphere :g:`BCDE <0429.g1:( B C D E )>`, the polyhedral solid in the sphere :g:`BCDE <0429.g1:( B C D E )>` has to the polyhedral solid in the other sphere the :term:`ratio <ratio>` triplicate of that which the diameter of the sphere :g:`BCDE <0429.g1:( B C D E )>` has to the diameter of the other sphere.
For, the solids being divided into their pyramids similar in multitude and arrangement, the pyramids will be similar.
But similar pyramids are to one another in the triplicate ratio of their corresponding sides; [XII. 8, Por.]
therefore the pyramid of which the quadrilateral KBPS is the base, and the point A the vertex, has to the similarly arranged pyramid in the other sphere the ratio triplicate of that which the corresponding side has to the corresponding side, that is, of that which the radius AB of the sphere about A as centre has to the radius of the other sphere.
Similarly also each pyramid of those in the sphere about A as centre has to each similarly arranged pyramid of those in the other sphere the ratio triplicate of that which AB has to the radius of the other sphere.
And, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents; [V. 12]
hence the whole polyhedral solid in the sphere about A as centre has to the whole polyhedral solid in the other sphere the ratio triplicate of that which AB has to the radius of the other sphere, that is, of that which the diameter BD has to the diameter of the other sphere. 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.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"]; "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.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.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.20" [label="III.20", fillcolor="#222244", tooltip="III.20: In a circle the angle at the centre is double of the angle at the circumference, when the angles have the same circumference as base.", URL="../../../../volume_II/book_iii/propositions/20/index.html", target="_top"]; "III.21" [label="III.21", fillcolor="#222244", tooltip="III.21: In a circle the angles in the same segment are equal to one another.", URL="../../../../volume_II/book_iii/propositions/21/index.html", target="_top"]; "III.22" [label="III.22", fillcolor="#222244", tooltip="III.22: The opposite angles of quadrilaterals in circles are equal to two right angles.", URL="../../../../volume_II/book_iii/propositions/22/index.html", target="_top"]; "III.24" [label="III.24", fillcolor="#222244", tooltip="III.24: Similar segments of circles on equal straight lines are equal to one another.", URL="../../../../volume_II/book_iii/propositions/24/index.html", target="_top"]; "III.26" [label="III.26", fillcolor="#222244", tooltip="III.26: In equal circles equal angles stand on equal circumferences, whether they stand at the centres or at the circumferences.", URL="../../../../volume_II/book_iii/propositions/26/index.html", target="_top"]; "III.27" [label="III.27", fillcolor="#222244", tooltip="III.27: In equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centres or at the circumferences.", URL="../../../../volume_II/book_iii/propositions/27/index.html", target="_top"]; "III.31" [label="III.31", fillcolor="#222244", tooltip="III.31: In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; and further the angle…", URL="../../../../volume_II/book_iii/propositions/31/index.html", target="_top"]; "IV.def.3" [label="IV.def.3", fillcolor="#224422", tooltip="IV.def.3: A rectilineal figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle.", URL="../../../../volume_II/book_iv/definitions/index.html#iv-def-3", target="_top"]; "IV.def.4" [label="IV.def.4", fillcolor="#224422", tooltip="IV.def.4: A rectilineal figure is said to be circumscribed about a circle, when each side of the circumscribed figure touches the circumference of the circle.", URL="../../../../volume_II/book_iv/definitions/index.html#iv-def-4", target="_top"]; "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.10" [label="V.def.10", fillcolor="#224422", tooltip="V.def.10: When four magnitudes are < continuously > proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually…", URL="../../../../volume_II/book_v/definitions/index.html#v-def-10", 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.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.20" [label="V.20", fillcolor="#222244", tooltip="V.20: If there be three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first be greater than the third, the…", URL="../../../../volume_II/book_v/propositions/20/index.html", target="_top"]; "V.22" [label="V.22", fillcolor="#222244", tooltip="V.22: If there be any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, they will also be in the same ratio…", URL="../../../../volume_II/book_v/propositions/22/index.html", target="_top"]; "VI.def.1" [label="VI.def.1", fillcolor="#224422", tooltip="VI.def.1: Similar rectilineal figures are such as have their angles severally equal and the sides about the equal angles proportional.", URL="../../../../volume_II/book_vi/definitions/index.html#vi-def-1", target="_top"]; "VI.def.2" [label="VI.def.2", fillcolor="#224422", tooltip="VI.def.2: [2. Reciprocally related figures. See note.]", URL="../../../../volume_II/book_vi/definitions/index.html#vi-def-2", target="_top"]; "VI.def.4" [label="VI.def.4", fillcolor="#224422", tooltip="VI.def.4: The height of any figure is the perpendicular drawn from the vertex to the base.", URL="../../../../volume_II/book_vi/definitions/index.html#vi-def-4", target="_top"]; "VI.1" [label="VI.1", fillcolor="#222244", tooltip="VI.1: Triangles and parallelograms which are under the same height are to one another as their bases.", URL="../../../../volume_II/book_vi/propositions/01/index.html", target="_top"]; "VI.2" [label="VI.2", fillcolor="#222244", tooltip="VI.2: If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally; and, if the sides of the triangle be cut…", URL="../../../../volume_II/book_vi/propositions/02/index.html", target="_top"]; "VI.4" [label="VI.4", fillcolor="#222244", tooltip="VI.4: In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles.", URL="../../../../volume_II/book_vi/propositions/04/index.html", target="_top"]; "VI.6" [label="VI.6", fillcolor="#222244", tooltip="VI.6: If two triangles have one angle equal to one angle and the sides about the equal angles proportional, the triangles will be equiangular and will have those angles equal which the…", URL="../../../../volume_II/book_vi/propositions/06/index.html", target="_top"]; "VI.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.15" [label="VI.15", fillcolor="#222244", tooltip="VI.15: In equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally proportional; and those triangles which have one angle equal to one…", URL="../../../../volume_II/book_vi/propositions/15/index.html", target="_top"]; "VI.18" [label="VI.18", fillcolor="#222244", tooltip="VI.18: On a given straight line to describe a rectilinear figure similar and similarly situated to a given rectilinear figure.", URL="../../../../volume_II/book_vi/propositions/18/index.html", target="_top"]; "VI.19" [label="VI.19", fillcolor="#222244", tooltip="VI.19: Similar triangles are to one another in the duplicate ratio of the corresponding sides.", URL="../../../../volume_II/book_vi/propositions/19/index.html", target="_top"]; "VI.20" [label="VI.20", fillcolor="#222244", tooltip="VI.20: Similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio…", URL="../../../../volume_II/book_vi/propositions/20/index.html", target="_top"]; "VI.22" [label="VI.22", fillcolor="#222244", tooltip="VI.22: If four straight lines be proportional, the rectilineal figures similar and similarly described upon them will also be proportional; and, if the rectilineal figures similar and…", URL="../../../../volume_II/book_vi/propositions/22/index.html", target="_top"]; "VII.def.1" [label="VII.def.1", fillcolor="#224422", tooltip="VII.def.1: An unit is that by virtue of which each of the things that exist is called one.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-1", target="_top"]; "VII.def.2" [label="VII.def.2", fillcolor="#224422", tooltip="VII.def.2: A number is a multitude composed of units.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-2", target="_top"]; "VII.def.3" [label="VII.def.3", fillcolor="#224422", tooltip="VII.def.3: A number is a part of a number, the less of the greater, when it measures the greater.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-3", target="_top"]; "VII.def.5" [label="VII.def.5", fillcolor="#224422", tooltip="VII.def.5: The greater number is a multiple of the less when it is measured by the less.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-5", target="_top"]; "VII.def.6" [label="VII.def.6", fillcolor="#224422", tooltip="VII.def.6: An even number is that which is divisible into two equal parts.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-6", target="_top"]; "VII.def.15" [label="VII.def.15", fillcolor="#224422", tooltip="VII.def.15: A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.", URL="../../../../volume_II/book_vii/definitions/index.html#vii-def-15", target="_top"]; "X.1" [label="X.1", fillcolor="#222244", tooltip="X.1: Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and…", URL="../../../book_x/propositions/001/index.html", target="_top"]; "XI.def.1" [label="XI.def.1", fillcolor="#224422", tooltip="XI.def.1: A solid is that which has length, breadth, and depth.", URL="../../../book_xi/definitions/index.html#xi-def-1", target="_top"]; "XI.def.2" [label="XI.def.2", fillcolor="#224422", tooltip="XI.def.2: An extremity of a solid is a surface.", URL="../../../book_xi/definitions/index.html#xi-def-2", target="_top"]; "XI.def.3" [label="XI.def.3", fillcolor="#224422", tooltip="XI.def.3: A straight line is at right angles to a plane, when it makes right angles with all the straight lines which meet it and are in the plane.", URL="../../../book_xi/definitions/index.html#xi-def-3", target="_top"]; "XI.def.4" [label="XI.def.4", fillcolor="#224422", tooltip="XI.def.4: A plane is at right angles to a plane when the straight lines drawn, in one of the planes, at right angles to the common section of the planes are at right angles to the remaining…", URL="../../../book_xi/definitions/index.html#xi-def-4", target="_top"]; "XI.def.8" [label="XI.def.8", fillcolor="#224422", tooltip="XI.def.8: Parallel planes are those which do not meet.", URL="../../../book_xi/definitions/index.html#xi-def-8", target="_top"]; "XI.def.9" [label="XI.def.9", fillcolor="#224422", tooltip="XI.def.9: Similar solid figures are those contained by similar planes equal in multitude.", URL="../../../book_xi/definitions/index.html#xi-def-9", target="_top"]; "XI.def.10" [label="XI.def.10", fillcolor="#224422", tooltip="XI.def.10: Equal and similar solid figures are those contained by similar planes equal in multitude and in magnitude.", URL="../../../book_xi/definitions/index.html#xi-def-10", target="_top"]; "XI.def.12" [label="XI.def.12", fillcolor="#224422", tooltip="XI.def.12: A pyramid is a solid figure, contained by planes, which is constructed from one plane to one point.", URL="../../../book_xi/definitions/index.html#xi-def-12", target="_top"]; "XI.def.13" [label="XI.def.13", fillcolor="#224422", tooltip="XI.def.13: A prism is a solid figure contained by planes two of which, namely those which are opposite, are equal, similar and parallel, while the rest are parallelograms.", URL="../../../book_xi/definitions/index.html#xi-def-13", target="_top"]; "XI.def.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.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.1" [label="XI.1", fillcolor="#222244", tooltip="XI.1: A part of a straight line cannot be in the plane of reference and a part in a plane more elevated.", URL="../../../book_xi/propositions/01/index.html", target="_top"]; "XI.2" [label="XI.2", fillcolor="#222244", tooltip="XI.2: If two straight lines cut one another, they are in one plane, and every triangle is in one plane.", URL="../../../book_xi/propositions/02/index.html", target="_top"]; "XI.3" [label="XI.3", fillcolor="#222244", tooltip="XI.3: If two planes cut one another, their common section is a straight line.", URL="../../../book_xi/propositions/03/index.html", target="_top"]; "XI.4" [label="XI.4", fillcolor="#222244", tooltip="XI.4: If a straight line be set up at right angles to two straight lines which cut one another, at their common point of section, it will also be at right angles to the plane through…", URL="../../../book_xi/propositions/04/index.html", target="_top"]; "XI.5" [label="XI.5", fillcolor="#222244", tooltip="XI.5: If a straight line be set up at right angles to three straight lines which meet one another, at their common point of section, the three straight lines are in one plane.", URL="../../../book_xi/propositions/05/index.html", target="_top"]; "XI.6" [label="XI.6", fillcolor="#222244", tooltip="XI.6: If two straight lines be at right angles to the same plane, the straight lines will be parallel.", URL="../../../book_xi/propositions/06/index.html", target="_top"]; "XI.7" [label="XI.7", fillcolor="#222244", tooltip="XI.7: If two straight lines be parallel and points be taken at random on each of them, the straight line joining the points is in the same plane with the parallel straight lines.", URL="../../../book_xi/propositions/07/index.html", target="_top"]; "XI.8" [label="XI.8", fillcolor="#222244", tooltip="XI.8: If two straight lines be parallel, and one of them be at right angles to any plane, the remaining one will also be at right angles to the same plane.", URL="../../../book_xi/propositions/08/index.html", target="_top"]; "XI.9" [label="XI.9", fillcolor="#222244", tooltip="XI.9: Straight lines which are parallel to the same straight line and are not in the same plane with it are also parallel to one another.", URL="../../../book_xi/propositions/09/index.html", target="_top"]; "XI.10" [label="XI.10", fillcolor="#222244", tooltip="XI.10: If two straight lines meeting one another be parallel to two straight lines meeting one another not in the same plane, they will contain equal angles.", URL="../../../book_xi/propositions/10/index.html", target="_top"]; "XI.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.16" [label="XI.16", fillcolor="#222244", tooltip="XI.16: If two parallel planes be cut by any plane, their common sections are parallel.", URL="../../../book_xi/propositions/16/index.html", target="_top"]; "XI.18" [label="XI.18", fillcolor="#222244", tooltip="XI.18: If a straight line be at right angles to any plane, all the planes through it will also be at right angles to the same plane.", URL="../../../book_xi/propositions/18/index.html", target="_top"]; "XI.24" [label="XI.24", fillcolor="#222244", tooltip="XI.24: If a solid be contained by parallel planes, the opposite planes in it are equal and parallelogrammic.", URL="../../../book_xi/propositions/24/index.html", target="_top"]; "XI.25" [label="XI.25", fillcolor="#222244", tooltip="XI.25: If a parallelepipedal solid be cut by a plane which is parallel to the opposite planes, then, as the base is to the base, so will the solid be to the solid.", URL="../../../book_xi/propositions/25/index.html", target="_top"]; "XI.28" [label="XI.28", fillcolor="#222244", tooltip="XI.28: If a parallelepipedal solid be cut by a plane through the diagonals of the opposite planes, the solid will be bisected by the plane.", URL="../../../book_xi/propositions/28/index.html", target="_top"]; "XI.29" [label="XI.29", fillcolor="#222244", tooltip="XI.29: Parallelepipedal solids which are on the same base and of the same height, and in which the extremities of the sides which stand up are on the same straight lines, are equal to…", URL="../../../book_xi/propositions/29/index.html", target="_top"]; "XI.30" [label="XI.30", fillcolor="#222244", tooltip="XI.30: Parallelepipedal solids which are on the same base and of the same height, and in which the extremities of the sides which stand up are not on the same straight lines, are equal…", URL="../../../book_xi/propositions/30/index.html", target="_top"]; "XI.31" [label="XI.31", fillcolor="#222244", tooltip="XI.31: Parallelepipedal solids which are on equal bases and of the same height are equal to one another.", URL="../../../book_xi/propositions/31/index.html", target="_top"]; "XI.32" [label="XI.32", fillcolor="#222244", tooltip="XI.32: Parallelepipedal solids which are of the same height are to one another as their bases.", URL="../../../book_xi/propositions/32/index.html", target="_top"]; "XI.33" [label="XI.33", fillcolor="#222244", tooltip="XI.33: Similar parallelepipedal solids are to one another in the triplicate ratio of their corresponding sides.", URL="../../../book_xi/propositions/33/index.html", target="_top"]; "XI.39" [label="XI.39", fillcolor="#222244", tooltip="XI.39: If there be two prisms of equal height, and one have a parallelogram as base and the other a triangle, and if the parallelogram be double of the triangle, the prisms will be equal.", URL="../../../book_xi/propositions/39/index.html", target="_top"]; "XII.1" [label="XII.1", fillcolor="#222244", tooltip="XII.1: Similar polygons inscribed in circles are to one another as the squares on the diameters.", URL="../01/index.html", target="_top"]; "XII.2" [label="XII.2", fillcolor="#222244", tooltip="XII.2: Circles are to one another as the squares on the diameters.", URL="../02/index.html", target="_top"]; "XII.3" [label="XII.3", fillcolor="#222244", tooltip="XII.3: Any pyramid which has a triangular base is divided into two pyramids equal and similar to one another, similar to the whole and having triangular bases, and into two equal prisms…", URL="../03/index.html", target="_top"]; "XII.4" [label="XII.4", fillcolor="#222244", tooltip="XII.4: If there be two pyramids of the same height which have triangular bases, and each of them be divided into two pyramids equal to one another and similar to the whole, and into two…", URL="../04/index.html", target="_top"]; "XII.5" [label="XII.5", fillcolor="#222244", tooltip="XII.5: Pyramids which are of the same height and have triangular bases are to one another as the bases.", URL="../05/index.html", target="_top"]; "XII.7" [label="XII.7", fillcolor="#222244", tooltip="XII.7: Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases.", URL="../07/index.html", target="_top"]; "XII.8" [label="XII.8", fillcolor="#222244", tooltip="XII.8: Similar pyramids which have triangular bases are in the triplicate ratio of their corresponding sides.", URL="../08/index.html", target="_top"]; "XII.17" [label="XII.17", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", tooltip="XII.17: Given two spheres about the same centre, to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface.", 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"; "III.def.8" -> "I.def.1"; "III.10" -> "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"; "I.def.3" -> "I.def.2"; "I.def.4" -> "I.def.2"; "I.def.6" -> "I.def.2"; "I.def.8" -> "I.def.2"; "I.def.9" -> "I.def.2"; "VI.2" -> "I.def.2"; "I.def.4" -> "I.def.3"; "I.def.13" -> "I.def.3"; "I.2" -> "I.def.3"; "I.7" -> "I.def.3"; "I.33" -> "I.def.3"; "III.def.8" -> "I.def.3"; "XI.def.2" -> "I.def.3"; "XI.29" -> "I.def.3"; "XI.30" -> "I.def.3"; "I.def.7" -> "I.def.4"; "I.def.8" -> "I.def.4"; "I.def.10" -> "I.def.4"; "I.def.15" -> "I.def.4"; "I.def.17" -> "I.def.4"; "I.def.19" -> "I.def.4"; "I.def.23" -> "I.def.4"; "I.post.1" -> "I.def.4"; "I.post.2" -> "I.def.4"; "I.post.5" -> "I.def.4"; "I.1" -> "I.def.4"; "I.2" -> "I.def.4"; "I.3" -> "I.def.4"; "I.4" -> "I.def.4"; "I.5" -> "I.def.4"; "I.7" -> "I.def.4"; "I.8" -> "I.def.4"; "I.10" -> "I.def.4"; "I.11" -> "I.def.4"; "I.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"; "III.def.6" -> "I.def.4"; "III.def.8" -> "I.def.4"; "III.def.9" -> "I.def.4"; "III.def.10" -> "I.def.4"; "III.24" -> "I.def.4"; "VI.2" -> "I.def.4"; "VI.11" -> "I.def.4"; "VI.12" -> "I.def.4"; "VI.18" -> "I.def.4"; "VI.22" -> "I.def.4"; "XI.def.3" -> "I.def.4"; "XI.def.4" -> "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.10" -> "I.def.4"; "XI.11" -> "I.def.4"; "XI.18" -> "I.def.4"; "XI.29" -> "I.def.4"; "XI.30" -> "I.def.4"; "I.def.6" -> "I.def.5"; "I.def.7" -> "I.def.5"; "XI.def.17" -> "I.def.5"; "XII.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"; "XI.def.3" -> "I.def.7"; "XI.def.4" -> "I.def.7"; "XI.def.12" -> "I.def.7"; "XI.def.13" -> "I.def.7"; "XI.1" -> "I.def.7"; "XI.2" -> "I.def.7"; "XI.3" -> "I.def.7"; "XI.5" -> "I.def.7"; "XI.7" -> "I.def.7"; "XI.8" -> "I.def.7"; "XI.9" -> "I.def.7"; "XI.10" -> "I.def.7"; "XI.11" -> "I.def.7"; "XI.16" -> "I.def.7"; "XI.18" -> "I.def.7"; "XI.24" -> "I.def.7"; "XI.25" -> "I.def.7"; "XI.28" -> "I.def.7"; "I.def.9" -> "I.def.8"; "I.def.10" -> "I.def.8"; "I.def.11" -> "I.def.8"; "I.def.12" -> "I.def.8"; "I.def.21" -> "I.def.8"; "I.def.22" -> "I.def.8"; "I.post.5" -> "I.def.8"; "I.4" -> "I.def.8"; "I.5" -> "I.def.8"; "I.8" -> "I.def.8"; "I.13" -> "I.def.8"; "I.14" -> "I.def.8"; "I.15" -> "I.def.8"; "I.16" -> "I.def.8"; "I.17" -> "I.def.8"; "I.18" -> "I.def.8"; "I.19" -> "I.def.8"; "I.26" -> "I.def.8"; "I.27" -> "I.def.8"; "I.28" -> "I.def.8"; "I.29" -> "I.def.8"; "I.32" -> "I.def.8"; "I.34" -> "I.def.8"; "III.def.8" -> "I.def.8"; "III.def.9" -> "I.def.8"; "III.def.10" -> "I.def.8"; "III.def.11" -> "I.def.8"; "III.20" -> "I.def.8"; "III.21" -> "I.def.8"; "III.22" -> "I.def.8"; "III.26" -> "I.def.8"; "III.27" -> "I.def.8"; "III.31" -> "I.def.8"; "IV.def.3" -> "I.def.8"; "VI.def.1" -> "I.def.8"; "VI.4" -> "I.def.8"; "VI.6" -> "I.def.8"; "VI.15" -> "I.def.8"; "XI.10" -> "I.def.8"; "I.9" -> "I.def.9"; "I.23" -> "I.def.9"; "I.42" -> "I.def.9"; "I.44" -> "I.def.9"; "I.45" -> "I.def.9"; "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.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.def.4" -> "I.def.10"; "XI.4" -> "I.def.10"; "XI.5" -> "I.def.10"; "XI.8" -> "I.def.10"; "XI.11" -> "I.def.10"; "XI.18" -> "I.def.10"; "I.def.21" -> "I.def.11"; "I.def.21" -> "I.def.12"; "I.def.14" -> "I.def.13"; "I.def.15" -> "I.def.14"; "I.def.18" -> "I.def.14"; "I.4" -> "I.def.14"; "III.def.6" -> "I.def.14"; "III.def.10" -> "I.def.14"; "IV.def.3" -> "I.def.14"; "IV.def.4" -> "I.def.14"; "V.9" -> "I.def.14"; "V.10" -> "I.def.14"; "V.11" -> "I.def.14"; "VI.def.4" -> "I.def.14"; "VI.18" -> "I.def.14"; "XI.def.14" -> "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.6" -> "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.20" -> "I.def.15"; "III.21" -> "I.def.15"; "III.22" -> "I.def.15"; "III.31" -> "I.def.15"; "IV.def.3" -> "I.def.15"; "IV.def.4" -> "I.def.15"; "XI.1" -> "I.def.15"; "XII.2" -> "I.def.15"; "I.def.17" -> "I.def.16"; "I.def.18" -> "I.def.16"; "I.post.3" -> "I.def.16"; "III.def.10" -> "I.def.16"; "III.1" -> "I.def.16"; "III.5" -> "I.def.16"; "III.26" -> "I.def.16"; "III.27" -> "I.def.16"; "XI.def.17" -> "I.def.16"; "XI.1" -> "I.def.16"; "XII.17" -> "I.def.16"; "I.def.18" -> "I.def.17"; "I.34" -> "I.def.17"; "I.43" -> "I.def.17"; "III.def.1" -> "I.def.17"; "III.31" -> "I.def.17"; "XI.def.14" -> "I.def.17"; "XI.1" -> "I.def.17"; "XII.1" -> "I.def.17"; "XII.2" -> "I.def.17"; "III.31" -> "I.def.18"; "XI.def.14" -> "I.def.18"; "I.def.20" -> "I.def.19"; "I.def.21" -> "I.def.19"; "I.def.22" -> "I.def.19"; "I.4" -> "I.def.19"; "I.8" -> "I.def.19"; "I.16" -> "I.def.19"; "I.17" -> "I.def.19"; "I.18" -> "I.def.19"; "I.19" -> "I.def.19"; "I.20" -> "I.def.19"; "I.22" -> "I.def.19"; "I.26" -> "I.def.19"; "I.32" -> "I.def.19"; "I.37" -> "I.def.19"; "I.38" -> "I.def.19"; "I.39" -> "I.def.19"; "I.41" -> "I.def.19"; "I.42" -> "I.def.19"; "I.44" -> "I.def.19"; "I.45" -> "I.def.19"; "III.22" -> "I.def.19"; "IV.def.3" -> "I.def.19"; "IV.def.4" -> "I.def.19"; "VI.1" -> "I.def.19"; "VI.2" -> "I.def.19"; "VI.4" -> "I.def.19"; "VI.6" -> "I.def.19"; "VI.8" -> "I.def.19"; "VI.15" -> "I.def.19"; "VI.20" -> "I.def.19"; "VI.22" -> "I.def.19"; "XI.2" -> "I.def.19"; "XI.39" -> "I.def.19"; "I.def.22" -> "I.def.20"; "I.1" -> "I.def.20"; "I.5" -> "I.def.20"; "I.47" -> "I.def.21"; "VI.8" -> "I.def.21"; "I.35" -> "I.def.22"; "I.46" -> "I.def.22"; "I.47" -> "I.def.22"; "XII.1" -> "I.def.22"; "XII.2" -> "I.def.22"; "I.27" -> "I.def.23"; "I.28" -> "I.def.23"; "I.29" -> "I.def.23"; "I.30" -> "I.def.23"; "I.31" -> "I.def.23"; "I.33" -> "I.def.23"; "I.35" -> "I.def.23"; "I.36" -> "I.def.23"; "I.37" -> "I.def.23"; "I.38" -> "I.def.23"; "I.39" -> "I.def.23"; "I.41" -> "I.def.23"; "VI.2" -> "I.def.23"; "XI.def.13" -> "I.def.23"; "XI.6" -> "I.def.23"; "XI.7" -> "I.def.23"; "XI.8" -> "I.def.23"; "XI.9" -> "I.def.23"; "XI.10" -> "I.def.23"; "XI.16" -> "I.def.23"; "XI.25" -> "I.def.23"; "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"; "VI.6" -> "I.4"; "XI.4" -> "I.4"; "XI.6" -> "I.4"; "XI.24" -> "I.4"; "XI.29" -> "I.4"; "XII.3" -> "I.4"; "I.7" -> "I.5"; "I.19" -> "I.5"; "I.20" -> "I.5"; "III.20" -> "I.5"; "III.31" -> "I.5"; "I.8" -> "I.7"; "I.9" -> "I.8"; "I.11" -> "I.8"; "I.12" -> "I.8"; "I.23" -> "I.8"; "III.1" -> "I.8"; "XI.4" -> "I.8"; "XI.6" -> "I.8"; "XI.10" -> "I.8"; "XI.29" -> "I.8"; "I.10" -> "I.9"; "I.12" -> "I.10"; "I.16" -> "I.10"; "I.13" -> "I.11"; "I.46" -> "I.11"; "XI.11" -> "I.11"; "XI.18" -> "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.15" -> "I.14"; "I.16" -> "I.15"; "I.28" -> "I.15"; "I.29" -> "I.15"; "I.44" -> "I.15"; "XI.4" -> "I.15"; "I.18" -> "I.16"; "I.26" -> "I.16"; "I.27" -> "I.16"; "III.31" -> "I.17"; "VI.4" -> "I.17"; "I.19" -> "I.18"; "I.20" -> "I.19"; "I.22" -> "I.20"; "I.23" -> "I.22"; "I.31" -> "I.23"; "I.42" -> "I.23"; "III.27" -> "I.23"; "VI.6" -> "I.23"; "VI.18" -> "I.23"; "XI.31" -> "I.23"; "I.34" -> "I.26"; "XI.4" -> "I.26"; "XII.17" -> "I.26"; "I.28" -> "I.27"; "I.31" -> "I.27"; "I.33" -> "I.27"; "VI.4" -> "I.28"; "XI.6" -> "I.28"; "XI.18" -> "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"; "XI.8" -> "I.29"; "XII.3" -> "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"; "VI.11" -> "I.31"; "VI.12" -> "I.31"; "XI.11" -> "I.31"; "III.20" -> "I.32"; "III.22" -> "I.32"; "III.31" -> "I.32"; "VI.6" -> "I.32"; "VI.8" -> "I.32"; "VI.18" -> "I.32"; "VI.20" -> "I.32"; "XII.1" -> "I.32"; "I.36" -> "I.33"; "I.45" -> "I.33"; "XI.10" -> "I.33"; "XII.17" -> "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"; "XI.24" -> "I.34"; "XI.28" -> "I.34"; "XI.29" -> "I.34"; "XI.39" -> "I.34"; "XII.3" -> "I.34"; "XII.7" -> "I.34"; "I.36" -> "I.35"; "I.37" -> "I.35"; "XI.31" -> "I.35"; "I.38" -> "I.36"; "XI.29" -> "I.36"; "I.39" -> "I.37"; "I.41" -> "I.37"; "I.42" -> "I.38"; "VI.1" -> "I.38"; "VI.2" -> "I.38"; "VI.2" -> "I.39"; "I.42" -> "I.41"; "I.47" -> "I.41"; "VI.1" -> "I.41"; "I.44" -> "I.42"; "I.45" -> "I.42"; "I.44" -> "I.43"; "I.45" -> "I.44"; "XI.32" -> "I.45"; "I.47" -> "I.46"; "XII.17" -> "I.47"; "XII.17" -> "II.def.1"; "III.26" -> "III.def.1"; "III.27" -> "III.def.1"; "XII.2" -> "III.def.6"; "III.31" -> "III.def.8"; "III.26" -> "III.def.9"; "III.27" -> "III.def.9"; "III.20" -> "III.def.10"; "III.24" -> "III.def.11"; "III.26" -> "III.def.11"; "III.10" -> "III.1"; "III.10" -> "III.5"; "III.24" -> "III.10"; "III.21" -> "III.20"; "III.27" -> "III.20"; "III.22" -> "III.21"; "III.31" -> "III.22"; "III.26" -> "III.24"; "III.27" -> "III.26"; "XII.1" -> "III.27"; "XII.17" -> "III.27"; "XII.1" -> "III.31"; "XII.17" -> "III.31"; "XII.1" -> "IV.def.3"; "XII.2" -> "IV.def.4"; "V.def.2" -> "V.def.1"; "VII.def.3" -> "V.def.1"; "VII.def.5" -> "V.def.1"; "V.def.5" -> "V.def.2"; "V.def.7" -> "V.def.2"; "V.1" -> "V.def.2"; "V.2" -> "V.def.2"; "V.3" -> "V.def.2"; "V.4" -> "V.def.2"; "V.15" -> "V.def.2"; "V.def.17" -> "V.def.3"; "V.10" -> "V.def.3"; "V.11" -> "V.def.3"; "VI.20" -> "V.def.3"; "V.8" -> "V.def.4"; "V.10" -> "V.def.4"; "X.1" -> "V.def.4"; "V.def.6" -> "V.def.5"; "V.4" -> "V.def.5"; "V.7" -> "V.def.5"; "V.9" -> "V.def.5"; "V.11" -> "V.def.5"; "V.12" -> "V.def.5"; "V.13" -> "V.def.5"; "V.14" -> "V.def.5"; "V.15" -> "V.def.5"; "V.16" -> "V.def.5"; "V.20" -> "V.def.5"; "V.22" -> "V.def.5"; "VI.1" -> "V.def.5"; "VI.20" -> "V.def.5"; "XI.25" -> "V.def.5"; "V.def.9" -> "V.def.6"; "V.def.10" -> "V.def.6"; "V.7" -> "V.def.6"; "V.12" -> "V.def.6"; "V.16" -> "V.def.6"; "VI.def.1" -> "V.def.6"; "VI.4" -> "V.def.6"; "VI.6" -> "V.def.6"; "VI.11" -> "V.def.6"; "VI.12" -> "V.def.6"; "VI.22" -> "V.def.6"; "V.8" -> "V.def.7"; "V.10" -> "V.def.7"; "V.13" -> "V.def.7"; "VI.19" -> "V.def.9"; "XI.33" -> "V.def.10"; "XII.8" -> "V.def.10"; "V.def.12" -> "V.def.11"; "V.def.13" -> "V.def.11"; "V.12" -> "V.def.11"; "VI.19" -> "V.def.11"; "V.16" -> "V.def.12"; "V.7" -> "V.def.13"; "V.3" -> "V.def.17"; "V.20" -> "V.def.17"; "V.22" -> "V.def.17"; "V.8" -> "V.1"; "V.12" -> "V.1"; "V.3" -> "V.2"; "V.4" -> "V.3"; "V.22" -> "V.4"; "V.10" -> "V.7"; "V.15" -> "V.7"; "VI.2" -> "V.7"; "VI.15" -> "V.7"; "XI.31" -> "V.7"; "V.9" -> "V.8"; "V.10" -> "V.8"; "V.14" -> "V.8"; "V.20" -> "V.8"; "VI.2" -> "V.9"; "VI.6" -> "V.9"; "VI.15" -> "V.9"; "VI.22" -> "V.9"; "XI.31" -> "V.9"; "V.14" -> "V.10"; "V.20" -> "V.10"; "V.16" -> "V.11"; "VI.1" -> "V.11"; "VI.2" -> "V.11"; "VI.6" -> "V.11"; "VI.15" -> "V.11"; "VI.19" -> "V.11"; "VI.22" -> "V.11"; "XI.31" -> "V.11"; "XII.2" -> "V.11"; "XII.5" -> "V.11"; "V.15" -> "V.12"; "VI.20" -> "V.12"; "XII.4" -> "V.12"; "XII.8" -> "V.12"; "XII.17" -> "V.12"; "V.14" -> "V.13"; "V.20" -> "V.13"; "V.16" -> "V.14"; "V.16" -> "V.15"; "VI.1" -> "V.15"; "VI.4" -> "V.16"; "VI.19" -> "V.16"; "XII.2" -> "V.16"; "XII.4" -> "V.16"; "XII.5" -> "V.16"; "V.22" -> "V.20"; "VI.4" -> "V.22"; "VI.20" -> "V.22"; "VI.22" -> "V.22"; "VI.8" -> "VI.def.1"; "VI.18" -> "VI.def.1"; "VI.19" -> "VI.def.1"; "VI.20" -> "VI.def.1"; "VI.22" -> "VI.def.1"; "XI.33" -> "VI.def.1"; "XII.1" -> "VI.def.1"; "XII.3" -> "VI.def.1"; "VI.15" -> "VI.def.2"; "VI.1" -> "VI.def.4"; "XI.29" -> "VI.def.4"; "XI.30" -> "VI.def.4"; "XI.31" -> "VI.def.4"; "XI.32" -> "VI.def.4"; "XI.39" -> "VI.def.4"; "XII.4" -> "VI.def.4"; "XII.5" -> "VI.def.4"; "VI.2" -> "VI.1"; "VI.15" -> "VI.1"; "VI.19" -> "VI.1"; "VI.20" -> "VI.1"; "XI.33" -> "VI.1"; "VI.4" -> "VI.2"; "VI.11" -> "VI.2"; "VI.12" -> "VI.2"; "XII.3" -> "VI.2"; "XII.17" -> "VI.2"; "VI.6" -> "VI.4"; "VI.8" -> "VI.4"; "VI.18" -> "VI.4"; "VI.20" -> "VI.4"; "XII.1" -> "VI.4"; "VI.20" -> "VI.6"; "XII.1" -> "VI.6"; "XII.17" -> "VI.8"; "VI.19" -> "VI.11"; "VI.22" -> "VI.11"; "VI.22" -> "VI.12"; "VI.19" -> "VI.15"; "VI.22" -> "VI.18"; "VI.20" -> "VI.19"; "VI.22" -> "VI.19"; "XII.1" -> "VI.20"; "XII.8" -> "VI.20"; "XII.4" -> "VI.22"; "VII.def.2" -> "VII.def.1"; "VII.def.15" -> "VII.def.1"; "VII.def.3" -> "VII.def.2"; "VII.def.5" -> "VII.def.2"; "VII.def.15" -> "VII.def.2"; "XI.1" -> "VII.def.3"; "X.1" -> "VII.def.5"; "XI.25" -> "VII.def.5"; "X.1" -> "VII.def.6"; "X.1" -> "VII.def.15"; "XII.5" -> "X.1"; "XI.def.2" -> "XI.def.1"; "XI.def.12" -> "XI.def.1"; "XI.def.13" -> "XI.def.1"; "XI.24" -> "XI.def.1"; "XI.25" -> "XI.def.1"; "XI.28" -> "XI.def.1"; "XI.29" -> "XI.def.1"; "XI.30" -> "XI.def.1"; "XI.31" -> "XI.def.1"; "XI.32" -> "XI.def.1"; "XI.33" -> "XI.def.1"; "XII.17" -> "XI.def.1"; "XI.def.9" -> "XI.def.2"; "XI.def.10" -> "XI.def.2"; "XI.4" -> "XI.def.3"; "XI.5" -> "XI.def.3"; "XI.6" -> "XI.def.3"; "XI.8" -> "XI.def.3"; "XI.11" -> "XI.def.3"; "XI.18" -> "XI.def.3"; "XII.17" -> "XI.def.3"; "XI.18" -> "XI.def.4"; "XII.17" -> "XI.def.4"; "XI.16" -> "XI.def.8"; "XI.24" -> "XI.def.8"; "XII.8" -> "XI.def.9"; "XI.28" -> "XI.def.10"; "XI.31" -> "XI.def.10"; "XI.33" -> "XI.def.10"; "XII.3" -> "XI.def.10"; "XII.3" -> "XI.def.12"; "XII.4" -> "XI.def.12"; "XII.5" -> "XI.def.12"; "XII.7" -> "XI.def.12"; "XI.28" -> "XI.def.13"; "XI.29" -> "XI.def.13"; "XI.39" -> "XI.def.13"; "XII.3" -> "XI.def.13"; "XII.4" -> "XI.def.13"; "XII.7" -> "XI.def.13"; "XI.def.17" -> "XI.def.14"; "XII.17" -> "XI.def.14"; "XII.17" -> "XI.def.17"; "XI.2" -> "XI.1"; "XI.16" -> "XI.1"; "XI.6" -> "XI.2"; "XI.8" -> "XI.2"; "XII.17" -> "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"; "XII.17" -> "XI.6"; "XI.8" -> "XI.7"; "XII.17" -> "XI.7"; "XI.9" -> "XI.8"; "XI.11" -> "XI.8"; "XI.18" -> "XI.8"; "XI.10" -> "XI.9"; "XII.17" -> "XI.9"; "XI.24" -> "XI.10"; "XII.3" -> "XI.10"; "XII.17" -> "XI.11"; "XI.24" -> "XI.16"; "XII.17" -> "XI.18"; "XI.25" -> "XI.24"; "XI.31" -> "XI.24"; "XI.33" -> "XI.24"; "XII.8" -> "XI.24"; "XI.31" -> "XI.25"; "XI.32" -> "XI.25"; "XI.39" -> "XI.28"; "XII.8" -> "XI.28"; "XI.30" -> "XI.29"; "XI.31" -> "XI.29"; "XI.31" -> "XI.30"; "XI.32" -> "XI.31"; "XI.39" -> "XI.31"; "XI.33" -> "XI.32"; "XII.8" -> "XI.33"; "XII.3" -> "XI.39"; "XII.4" -> "XI.39"; "XII.2" -> "XII.1"; "XII.5" -> "XII.2"; "XII.4" -> "XII.3"; "XII.5" -> "XII.3"; "XII.5" -> "XII.4"; "XII.7" -> "XII.5"; "XII.8" -> "XII.7"; "XII.17" -> "XII.8"; }

Required for

XII.18

See Commentary on Proposition XII.17.