PROPOSITION 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.

Let A, B be two commensurable magnitudes, and let one of them, A, be incommensurable with any other magnitude C; I say that the remaining one, B, will also be incommensurable with C.

A C B
For, if B is commensurable with C, while A is also commensurable with B,
[X. 12]A is also commensurable with C.
But it is also incommensurable with it :
which is impossible.
Therefore B is not commensurable with C; therefore it is incommensurable with it.
Therefore etc.

LEMMA.

Given two unequal straight lines, to find by what square the square on the greater is greater than the square on the less.

Let AB, C be the given two unequal straight lines, and let AB be the greater of them; thus it is required to find by what square the square on AB is greater than the square on C.

A D B C

Let the semicircle ADB be described on AB, and let AD be fitted into it equal to C; [IV. 1] let DB be joined.

It is then manifest that the angle ADB is right, [III. 31] and that the square on AB is greater than the square on AD, that is, C, by the square on DB. [I. 47]

Similarly also, if two straight lines be given, the straight line the square on which is equal to the sum of the squares on them is found in this manner.

Let AD, DB be the given two straight lines, and let it be required to find the straight line the square on which is equal to the sum of the squares on them.

Let them be placed so as to contain a right angle, that formed by AD, DB; and let AB be joined.

It is again manifest that the straight line the square on which is equal to the sum of the squares on AD, DB is AB. [I. 47]
Q. E. D.

Dependency Graph

digraph { bgcolor="transparent"; node [shape=box, style="rounded,filled", fontname="Helvetica", fontsize=11, color="#d6d3cd", fontcolor="#ffffff"]; edge [color="#8c857b", fontcolor="#8c857b", arrowsize=0.75]; rankdir="TB"; "I.def.1" [label="I.def.1", fillcolor="#224422", tooltip="I.def.1: A point is that which has no part.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-1", target="_top"]; "I.def.2" [label="I.def.2", fillcolor="#224422", tooltip="I.def.2: A line is breadthless length.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-2", target="_top"]; "I.def.3" [label="I.def.3", fillcolor="#224422", tooltip="I.def.3: The extremities of a line are points.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-3", target="_top"]; "I.def.4" [label="I.def.4", fillcolor="#224422", tooltip="I.def.4: A straight line is a line which lies evenly with the points on itself.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-4", target="_top"]; "I.def.5" [label="I.def.5", fillcolor="#224422", tooltip="I.def.5: A surface is that which has length and breadth only.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-5", target="_top"]; "I.def.6" [label="I.def.6", fillcolor="#224422", tooltip="I.def.6: The extremities of a surface are lines.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-6", target="_top"]; "I.def.7" [label="I.def.7", fillcolor="#224422", tooltip="I.def.7: A plane surface is a surface which lies evenly with the straight lines on itself.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-7", target="_top"]; "I.def.8" [label="I.def.8", fillcolor="#224422", tooltip="I.def.8: A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-8", target="_top"]; "I.def.9" [label="I.def.9", fillcolor="#224422", tooltip="I.def.9: And when the lines containing the angle are straight, the angle is called rectilineal.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-9", target="_top"]; "I.def.10" [label="I.def.10", fillcolor="#224422", tooltip="I.def.10: When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-10", target="_top"]; "I.def.11" [label="I.def.11", fillcolor="#224422", tooltip="I.def.11: An obtuse angle is an angle greater than a right angle.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-11", target="_top"]; "I.def.12" [label="I.def.12", fillcolor="#224422", tooltip="I.def.12: An acute angle is an angle less than a right angle.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-12", target="_top"]; "I.def.13" [label="I.def.13", fillcolor="#224422", tooltip="I.def.13: A boundary is that which is an extremity of anything.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-13", target="_top"]; "I.def.14" [label="I.def.14", fillcolor="#224422", tooltip="I.def.14: A figure is that which is contained by any boundary or boundaries.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-14", target="_top"]; "I.def.15" [label="I.def.15", fillcolor="#224422", tooltip="I.def.15: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-15", target="_top"]; "I.def.16" [label="I.def.16", fillcolor="#224422", tooltip="I.def.16: And the point is called the centre of the circle.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-16", target="_top"]; "I.def.17" [label="I.def.17", fillcolor="#224422", tooltip="I.def.17: A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-17", target="_top"]; "I.def.18" [label="I.def.18", fillcolor="#224422", tooltip="I.def.18: A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-18", target="_top"]; "I.def.19" [label="I.def.19", fillcolor="#224422", tooltip="I.def.19: Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-19", target="_top"]; "I.def.20" [label="I.def.20", fillcolor="#224422", tooltip="I.def.20: Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-20", target="_top"]; "I.def.21" [label="I.def.21", fillcolor="#224422", tooltip="I.def.21: Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-21", target="_top"]; "I.def.22" [label="I.def.22", fillcolor="#224422", tooltip="I.def.22: Of quadrilateral figures, a square is that which is both equilateral and right-angled ; an oblong that which is right-angled but not equilateral ; a rhombus that which is…", URL="../../../../volume_I/book_i/definitions/index.html#i-def-22", target="_top"]; "I.def.23" [label="I.def.23", fillcolor="#224422", tooltip="I.def.23: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.", URL="../../../../volume_I/book_i/definitions/index.html#i-def-23", target="_top"]; "I.post.1" [label="I.post.1", fillcolor="#444422", tooltip="I.post.1: To draw a straight line from any point to any point.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-1", target="_top"]; "I.post.2" [label="I.post.2", fillcolor="#444422", tooltip="I.post.2: To produce a finite straight line continuously in a straight line.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-2", target="_top"]; "I.post.3" [label="I.post.3", fillcolor="#444422", tooltip="I.post.3: To describe a circle with any centre and distance.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-3", target="_top"]; "I.post.4" [label="I.post.4", fillcolor="#444422", tooltip="I.post.4: That all right angles are equal to one another.", URL="../../../../volume_I/book_i/postulates/index.html#i-post-4", target="_top"]; "I.post.5" [label="I.post.5", fillcolor="#444422", tooltip="I.post.5: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet…", URL="../../../../volume_I/book_i/postulates/index.html#i-post-5", target="_top"]; "I.cn.1" [label="I.cn.1", fillcolor="#442222", tooltip="I.cn.1: Things which are equal to the same thing are also equal to one another.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-1", target="_top"]; "I.cn.2" [label="I.cn.2", fillcolor="#442222", tooltip="I.cn.2: If equals be added to equals, the wholes are equal.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-2", target="_top"]; "I.cn.3" [label="I.cn.3", fillcolor="#442222", tooltip="I.cn.3: If equals be subtracted from equals, the remainders are equal.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-3", target="_top"]; "I.cn.4" [label="I.cn.4", fillcolor="#442222", tooltip="I.cn.4: Things which coincide with one another are equal to one another.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-4", target="_top"]; "I.cn.5" [label="I.cn.5", fillcolor="#442222", tooltip="I.cn.5: The whole is greater than the part.", URL="../../../../volume_I/book_i/common_notions/index.html#i-cn-5", target="_top"]; "I.1" [label="I.1", fillcolor="#222244", tooltip="I.1: On a given finite straight line to construct an equilateral triangle.", URL="../../../../volume_I/book_i/propositions/01/index.html", target="_top"]; "I.2" [label="I.2", fillcolor="#222244", tooltip="I.2: To place at a given point (as an extremity) a straight line equal to a given straight line.", URL="../../../../volume_I/book_i/propositions/02/index.html", target="_top"]; "I.3" [label="I.3", fillcolor="#222244", tooltip="I.3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less.", URL="../../../../volume_I/book_i/propositions/03/index.html", target="_top"]; "I.4" [label="I.4", fillcolor="#222244", tooltip="I.4: If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the…", URL="../../../../volume_I/book_i/propositions/04/index.html", target="_top"]; "I.5" [label="I.5", fillcolor="#222244", tooltip="I.5: In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one…", URL="../../../../volume_I/book_i/propositions/05/index.html", target="_top"]; "I.7" [label="I.7", fillcolor="#222244", tooltip="I.7: Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be constructed on the same straight line (from its…", URL="../../../../volume_I/book_i/propositions/07/index.html", target="_top"]; "I.8" [label="I.8", fillcolor="#222244", tooltip="I.8: If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the…", URL="../../../../volume_I/book_i/propositions/08/index.html", target="_top"]; "I.9" [label="I.9", fillcolor="#222244", tooltip="I.9: To bisect a given rectilineal angle.", URL="../../../../volume_I/book_i/propositions/09/index.html", target="_top"]; "I.10" [label="I.10", fillcolor="#222244", tooltip="I.10: To bisect a given finite straight line.", URL="../../../../volume_I/book_i/propositions/10/index.html", target="_top"]; "I.11" [label="I.11", fillcolor="#222244", tooltip="I.11: To draw a straight line at right angles to a given straight line from a given point on it.", URL="../../../../volume_I/book_i/propositions/11/index.html", target="_top"]; "I.13" [label="I.13", fillcolor="#222244", tooltip="I.13: If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles.", URL="../../../../volume_I/book_i/propositions/13/index.html", target="_top"]; "I.14" [label="I.14", fillcolor="#222244", tooltip="I.14: If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be…", URL="../../../../volume_I/book_i/propositions/14/index.html", target="_top"]; "I.15" [label="I.15", fillcolor="#222244", tooltip="I.15: If two straight lines cut one another, they make the vertical angles equal to one another.", URL="../../../../volume_I/book_i/propositions/15/index.html", target="_top"]; "I.16" [label="I.16", fillcolor="#222244", tooltip="I.16: In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles.", URL="../../../../volume_I/book_i/propositions/16/index.html", target="_top"]; "I.17" [label="I.17", fillcolor="#222244", tooltip="I.17: In any triangle two angles taken together in any manner are less than two right angles.", URL="../../../../volume_I/book_i/propositions/17/index.html", target="_top"]; "I.18" [label="I.18", fillcolor="#222244", tooltip="I.18: In any triangle the greater side subtends the greater angle.", URL="../../../../volume_I/book_i/propositions/18/index.html", target="_top"]; "I.19" [label="I.19", fillcolor="#222244", tooltip="I.19: In any triangle the greater angle is subtended by the greater side.", URL="../../../../volume_I/book_i/propositions/19/index.html", target="_top"]; "I.20" [label="I.20", fillcolor="#222244", tooltip="I.20: In any triangle two sides taken together in any manner are greater than the remaining one.", URL="../../../../volume_I/book_i/propositions/20/index.html", target="_top"]; "I.22" [label="I.22", fillcolor="#222244", tooltip="I.22: Out of three straight lines, which are equal to three given straight lines, to construct a triangle: thus it is necessary that two of the straight lines taken together in any…", URL="../../../../volume_I/book_i/propositions/22/index.html", target="_top"]; "I.23" [label="I.23", fillcolor="#222244", tooltip="I.23: On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.", URL="../../../../volume_I/book_i/propositions/23/index.html", target="_top"]; "I.26" [label="I.26", fillcolor="#222244", tooltip="I.26: If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one…", URL="../../../../volume_I/book_i/propositions/26/index.html", target="_top"]; "I.27" [label="I.27", fillcolor="#222244", tooltip="I.27: If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.", URL="../../../../volume_I/book_i/propositions/27/index.html", target="_top"]; "I.29" [label="I.29", fillcolor="#222244", tooltip="I.29: A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior…", URL="../../../../volume_I/book_i/propositions/29/index.html", target="_top"]; "I.31" [label="I.31", fillcolor="#222244", tooltip="I.31: Through a given point to draw a straight line parallel to a given straight line.", URL="../../../../volume_I/book_i/propositions/31/index.html", target="_top"]; "I.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.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"]; "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.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.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.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.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"]; "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.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.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.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.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"]; "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.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.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.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.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.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"]; "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="../../definitions/index.html#x-def-1", 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="../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="../006/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="../012/index.html", target="_top"]; "X.13" [label="X.13", penwidth=2.5, color="#e8b85a", fillcolor="#4a453b", 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="index.html", target="_top"]; "I.def.3" -> "I.def.1"; "I.def.4" -> "I.def.1"; "I.def.15" -> "I.def.1"; "I.def.16" -> "I.def.1"; "I.post.1" -> "I.def.1"; "I.2" -> "I.def.1"; "I.7" -> "I.def.1"; "I.11" -> "I.def.1"; "I.14" -> "I.def.1"; "I.23" -> "I.def.1"; "I.31" -> "I.def.1"; "III.def.8" -> "I.def.1"; "VI.2" -> "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"; "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.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.29" -> "I.def.4"; "I.31" -> "I.def.4"; "I.33" -> "I.def.4"; "I.46" -> "I.def.4"; "III.def.8" -> "I.def.4"; "III.def.10" -> "I.def.4"; "IV.def.7" -> "I.def.4"; "IV.1" -> "I.def.4"; "VI.2" -> "I.def.4"; "VI.11" -> "I.def.4"; "I.def.6" -> "I.def.5"; "I.def.7" -> "I.def.5"; "I.def.15" -> "I.def.6"; "I.def.8" -> "I.def.7"; "I.def.15" -> "I.def.7"; "I.def.23" -> "I.def.7"; "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.29" -> "I.def.8"; "I.32" -> "I.def.8"; "I.34" -> "I.def.8"; "III.def.8" -> "I.def.8"; "III.def.10" -> "I.def.8"; "III.20" -> "I.def.8"; "III.21" -> "I.def.8"; "III.22" -> "I.def.8"; "III.31" -> "I.def.8"; "VI.def.1" -> "I.def.8"; "VI.15" -> "I.def.8"; "I.9" -> "I.def.9"; "I.23" -> "I.def.9"; "I.def.11" -> "I.def.10"; "I.def.12" -> "I.def.10"; "I.def.21" -> "I.def.10"; "I.post.4" -> "I.def.10"; "I.post.5" -> "I.def.10"; "I.11" -> "I.def.10"; "I.13" -> "I.def.10"; "I.14" -> "I.def.10"; "I.17" -> "I.def.10"; "I.29" -> "I.def.10"; "I.32" -> "I.def.10"; "I.47" -> "I.def.10"; "III.22" -> "I.def.10"; "III.31" -> "I.def.10"; "VI.def.4" -> "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.10" -> "I.def.14"; "V.9" -> "I.def.14"; "V.10" -> "I.def.14"; "V.11" -> "I.def.14"; "VI.def.4" -> "I.def.14"; "I.def.16" -> "I.def.15"; "I.def.17" -> "I.def.15"; "I.def.18" -> "I.def.15"; "I.post.3" -> "I.def.15"; "I.1" -> "I.def.15"; "I.3" -> "I.def.15"; "III.def.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.7" -> "I.def.15"; "IV.1" -> "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"; "I.def.18" -> "I.def.17"; "I.34" -> "I.def.17"; "III.31" -> "I.def.17"; "IV.1" -> "I.def.17"; "III.31" -> "I.def.18"; "I.def.20" -> "I.def.19"; "I.def.21" -> "I.def.19"; "I.def.22" -> "I.def.19"; "I.4" -> "I.def.19"; "I.8" -> "I.def.19"; "I.16" -> "I.def.19"; "I.17" -> "I.def.19"; "I.18" -> "I.def.19"; "I.19" -> "I.def.19"; "I.20" -> "I.def.19"; "I.22" -> "I.def.19"; "I.26" -> "I.def.19"; "I.32" -> "I.def.19"; "I.37" -> "I.def.19"; "I.38" -> "I.def.19"; "I.39" -> "I.def.19"; "I.41" -> "I.def.19"; "III.22" -> "I.def.19"; "VI.1" -> "I.def.19"; "VI.2" -> "I.def.19"; "VI.15" -> "I.def.19"; "I.def.22" -> "I.def.20"; "I.1" -> "I.def.20"; "I.5" -> "I.def.20"; "I.47" -> "I.def.21"; "I.35" -> "I.def.22"; "I.46" -> "I.def.22"; "I.47" -> "I.def.22"; "I.27" -> "I.def.23"; "I.29" -> "I.def.23"; "I.31" -> "I.def.23"; "I.33" -> "I.def.23"; "I.35" -> "I.def.23"; "I.36" -> "I.def.23"; "I.37" -> "I.def.23"; "I.38" -> "I.def.23"; "I.39" -> "I.def.23"; "I.41" -> "I.def.23"; "VI.2" -> "I.def.23"; "I.1" -> "I.post.1"; "I.2" -> "I.post.1"; "I.5" -> "I.post.1"; "I.7" -> "I.post.1"; "I.16" -> "I.post.1"; "I.2" -> "I.post.2"; "I.5" -> "I.post.2"; "I.16" -> "I.post.2"; "I.17" -> "I.post.2"; "I.1" -> "I.post.3"; "I.2" -> "I.post.3"; "I.3" -> "I.post.3"; "I.14" -> "I.post.4"; "I.15" -> "I.post.4"; "I.29" -> "I.post.5"; "I.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.35" -> "I.cn.1"; "I.36" -> "I.cn.1"; "I.39" -> "I.cn.1"; "I.13" -> "I.cn.2"; "I.29" -> "I.cn.2"; "I.34" -> "I.cn.2"; "I.35" -> "I.cn.2"; "I.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.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"; "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.23" -> "I.8"; "I.10" -> "I.9"; "I.16" -> "I.10"; "I.13" -> "I.11"; "I.46" -> "I.11"; "I.14" -> "I.13"; "I.15" -> "I.13"; "I.17" -> "I.13"; "I.29" -> "I.13"; "I.32" -> "I.13"; "I.47" -> "I.14"; "VI.15" -> "I.14"; "I.16" -> "I.15"; "I.29" -> "I.15"; "I.18" -> "I.16"; "I.26" -> "I.16"; "I.27" -> "I.16"; "III.31" -> "I.17"; "I.19" -> "I.18"; "I.20" -> "I.19"; "I.22" -> "I.20"; "I.23" -> "I.22"; "I.31" -> "I.23"; "I.34" -> "I.26"; "I.31" -> "I.27"; "I.33" -> "I.27"; "I.32" -> "I.29"; "I.33" -> "I.29"; "I.34" -> "I.29"; "I.35" -> "I.29"; "I.46" -> "I.29"; "I.32" -> "I.31"; "I.37" -> "I.31"; "I.38" -> "I.31"; "I.39" -> "I.31"; "I.46" -> "I.31"; "VI.11" -> "I.31"; "III.20" -> "I.32"; "III.22" -> "I.32"; "III.31" -> "I.32"; "I.36" -> "I.33"; "I.35" -> "I.34"; "I.36" -> "I.34"; "I.37" -> "I.34"; "I.38" -> "I.34"; "I.41" -> "I.34"; "I.46" -> "I.34"; "I.36" -> "I.35"; "I.37" -> "I.35"; "I.38" -> "I.36"; "I.39" -> "I.37"; "I.41" -> "I.37"; "VI.1" -> "I.38"; "VI.2" -> "I.38"; "VI.2" -> "I.39"; "I.47" -> "I.41"; "VI.1" -> "I.41"; "I.47" -> "I.46"; "X.13" -> "I.47"; "III.31" -> "III.def.8"; "III.20" -> "III.def.10"; "III.21" -> "III.20"; "III.22" -> "III.21"; "III.31" -> "III.22"; "X.13" -> "III.31"; "IV.1" -> "IV.def.7"; "X.13" -> "IV.1"; "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.34" -> "V.def.1"; "VII.35" -> "V.def.1"; "X.def.1" -> "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"; "VII.33" -> "V.def.3"; "VIII.4" -> "V.def.3"; "X.5" -> "V.def.3"; "X.6" -> "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"; "VII.17" -> "V.def.5"; "VII.18" -> "V.def.5"; "VII.20" -> "V.def.5"; "VII.21" -> "V.def.5"; "VII.33" -> "V.def.5"; "V.def.9" -> "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.11" -> "V.def.6"; "VII.13" -> "V.def.6"; "VII.19" -> "V.def.6"; "VIII.4" -> "V.def.6"; "V.8" -> "V.def.7"; "V.10" -> "V.def.7"; "V.13" -> "V.def.7"; "VI.19" -> "V.def.9"; "V.def.12" -> "V.def.11"; "V.def.13" -> "V.def.11"; "V.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.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"; "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.15" -> "V.9"; "VII.19" -> "V.9"; "X.6" -> "V.9"; "V.14" -> "V.10"; "V.20" -> "V.10"; "V.16" -> "V.11"; "VI.1" -> "V.11"; "VI.2" -> "V.11"; "VI.15" -> "V.11"; "VI.19" -> "V.11"; "X.6" -> "V.11"; "X.12" -> "V.11"; "V.15" -> "V.12"; "V.14" -> "V.13"; "V.20" -> "V.13"; "V.16" -> "V.14"; "V.16" -> "V.15"; "VI.1" -> "V.15"; "VI.19" -> "V.16"; "V.22" -> "V.20"; "X.5" -> "V.22"; "X.6" -> "V.22"; "X.12" -> "V.22"; "VI.19" -> "VI.def.1"; "VI.15" -> "VI.def.2"; "VI.1" -> "VI.def.4"; "VI.2" -> "VI.1"; "VI.15" -> "VI.1"; "VI.19" -> "VI.1"; "VI.11" -> "VI.2"; "VI.19" -> "VI.11"; "VI.19" -> "VI.15"; "X.6" -> "VI.19"; "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.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.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.33" -> "VII.def.2"; "VII.34" -> "VII.def.2"; "VII.35" -> "VII.def.2"; "VIII.4" -> "VII.def.2"; "X.5" -> "VII.def.2"; "X.6" -> "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"; "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"; "VII.1" -> "VII.def.12"; "VII.2" -> "VII.def.12"; "VII.3" -> "VII.def.12"; "VII.21" -> "VII.def.12"; "VII.16" -> "VII.def.15"; "VII.17" -> "VII.def.15"; "VII.18" -> "VII.def.15"; "VII.33" -> "VII.def.15"; "VII.34" -> "VII.def.15"; "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"; "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.17" -> "VII.13"; "VII.20" -> "VII.13"; "VIII.4" -> "VII.13"; "VII.16" -> "VII.15"; "VII.18" -> "VII.16"; "VII.21" -> "VII.16"; "VII.33" -> "VII.16"; "VII.34" -> "VII.16"; "VII.18" -> "VII.17"; "VII.19" -> "VII.17"; "VII.34" -> "VII.17"; "VII.19" -> "VII.18"; "VII.33" -> "VII.19"; "VII.34" -> "VII.19"; "VII.21" -> "VII.20"; "VII.34" -> "VII.20"; "VIII.4" -> "VII.20"; "VII.33" -> "VII.21"; "VII.34" -> "VII.21"; "VII.34" -> "VII.33"; "VIII.4" -> "VII.34"; "VIII.4" -> "VII.35"; "VIII.4" -> "VIII.def.20"; "X.12" -> "VIII.4"; "X.5" -> "X.def.1"; "X.6" -> "X.def.1"; "X.12" -> "X.def.1"; "X.13" -> "X.def.1"; "X.12" -> "X.5"; "X.12" -> "X.6"; "X.13" -> "X.12"; }

Required for

X.18, X.22, X.23, X.24, X.25, X.26, X.27, X.28, X.31, X.32, X.33, X.34, X.35, X.36, X.37, X.38, X.39, X.40, X.41, X.42, X.43, X.44, X.45, X.46, X.47, X.48, X.49, X.50, X.52, X.53, X.54, X.55, X.56, X.57, X.58, X.59, X.60, X.61, X.62, X.63, X.64, X.65, X.66, X.67, X.68, X.69, X.70, X.71, X.72, X.73, X.75, X.76, X.77, X.78, X.79, X.80, X.81, X.82, X.83, X.84, X.85, X.86, X.87, X.88, X.89, X.90, X.91, X.92, X.93, X.94, X.95, X.96, X.97, X.98, X.99, X.100, X.101, X.102, X.103, X.104, X.105, X.106, X.107, X.108, X.109, X.110, X.111, X.112, X.113, X.114, XIII.6, XIII.11, XIII.16, XIII.17, XIII.18

See Commentary on Proposition X.13.