hcju/proofwikips huggingface dataset:
Definition:Urn
An '''urn''' is a hypothetical container from which one may draw one of a number of objects (usually defined to be balls) of assorted colours. The idea is that one cannot see the ball one is withdrawing until it is out of the urn. Therefore it is not possible to know in advance what colour ball one is taking. The probability that a particular ball being withdrawn has a particular colour depends entirely on the distribution within the urn of the various colours of the balls within. This is used as an abstraction of a randomizing device in probability theory.
Definition:Quadrilateral/Square
A '''square''' is a regular quadrilateral. That is, a regular polygon with $4$ sides. That is, a '''square''' is a plane figure with four sides all the same length and whose angles are all equal. :200px
Definition:Quadrilateral/Oblong
An '''oblong''' is a quadrilateral whose angles are all right angles, but whose sides are ''not'' all the same length: :400px
Definition:Quadrilateral/Rectangle/Containment
 :''{{:Definition:Euclid's Definitions - Book II/1 - Containment of Rectangle}}'' {{EuclidDefRefNocat|II|1|Containment of Rectangle}}
Definition:Quadrilateral/Rectangle
A '''rectangle''' is a quadrilateral all of whose angles are equal to a right angle, and whose sides ''may or may not'' all be the same length.
Definition:Quadrilateral/Parallelogram/Base
In a given parallelogram, one of the sides is distinguished as being the '''base'''. It is immaterial which is so chosen, but usual practice is that it is one of the two longer sides. In the parallelogram above, line $AB$ is considered to be the '''base'''.
Definition:Quadrilateral/Parallelogram/Altitude
An '''altitude''' of a parallelogram is a line drawn perpendicular to its base, through one of its vertices to the side opposite to the base (which is extended if necessary). In the parallelogram above, line $DE$ is an '''altitude''' of the parallelogram $ABCD$. The term is also used for the length of such a line.
Definition:Quadrilateral/Parallelogram
A '''parallelogram''' is a quadrilateral whose opposite sides are parallel to each other, and whose sides ''may or may not'' all be the same length.   :450px
Definition:Quadrilateral/Rhombus
A '''rhombus''' is a parallelogram whose sides are all the same length. :350px Its angles ''may or may not'' all be equal.
Definition:Quadrilateral/Rhomboid
A '''rhomboid''' is a parallelogram whose sides are ''not'' all the same length. Its angles ''may or may not'' all be equal.
Definition:Quadrilateral/Trapezoid
A '''trapezoid''' is a quadrilateral which has '''exactly one''' pair of sides parallel: :700px
Definition:Quadrilateral/Trapezium
A '''trapezium''' is a quadrilateral with no parallel sides. :300px
Definition:Quadrilateral/Kite
A '''kite''' is an irregular quadrilateral which has both pairs of adjacent sides equal. :300px
Definition:Quadrilateral/Dart
A '''dart''' is an irregular quadrilateral with a reflex angle. :500px
Definition:Boundary (Geometry)/Containment
A geometric figure is said to be '''contained''' by its boundary or boundaries.
Definition:Polygon/Side
The line segments which make up a polygon are known as its '''sides'''. Thus, in the polygon above, the '''sides''' are identified as $a, b, c, d$ and $e$.
Definition:Polygon/Vertex
A corner of a polygon is known as a '''vertex'''. Thus, in the polygon above, the vertices are $A, B, C, D$ and $E$.
Definition:Polygon/Adjacent
=== Adjacent Side to Vertex === {{:Definition:Polygon/Adjacent/Side to Vertex}} === Adjacent Sides === {{:Definition:Polygon/Adjacent/Sides}} === Adjacent Vertex to Side === {{:Definition:Polygon/Adjacent/Vertex to Side}} === Adjacent Vertices === {{:Definition:Polygon/Adjacent/Vertices}}
Definition:Polygon/Opposite
When a polygon has an even number of sides, each side has an '''opposite side''', and each vertex likewise has an '''opposite vertex'''. When a polygon has an odd number of sides, each side has an '''opposite vertex'''. The '''opposite side''' (or '''opposite vertex''') to a given side (or '''vertex''') is that side (or vertex) which has the same number of sides between it and the side (or vertex) in question.
Definition:Polygon/Internal Angle
The '''internal angle''' of a vertex of a polygon is the size of the angle between the sides adjacent to that vertex, as measured ''inside'' the polygon.
Definition:Polygon/External Angle
Contrary to intuition, the '''external angle''' of a vertex of a polygon is ''not'' the size of the angle between the sides forming that vertex, as measured outside the polygon. An '''external angle''' is in fact an angle formed by one side of a polygon and a line produced from an adjacent side. :200px While $\angle AFE$ is the internal angle of vertex $F$, the '''external angle''' of this vertex is $\angle EFG$.
Definition:Polygon/Base
For a given polygon, any one of its sides may be temporarily distinguished from the others, and referred to as the '''base'''. It is immaterial which is so chosen. The usual practice is that the polygon is drawn so that the '''base''' is made horizontal, and at the bottom.
Definition:Polygon/Equilateral
An '''equilateral polygon''' is a polygon in which all the sides are the same length.
Definition:Polygon/Equiangular
An '''equiangular polygon''' is a polygon in which all the vertices have the same angle.
Definition:Polygon/Regular
A '''regular polygon''' is a polygon which is both equilateral and equiangular. That is, in which all the sides are the same length, and all the vertices have the same angle: :300px
Definition:Polygon/Multi-lateral
A '''multi-lateral polygon''' is a term to define a polygon with more than four sides.
Definition:Triangle (Geometry)/Right-Angled/Hypotenuse
In a right-angled triangle, the opposite side to the right angle is known as the '''hypotenuse'''. In the above figure, the side labeled $b$ is the '''hypotenuse'''.
Definition:Triangle (Geometry)/Right-Angled/Adjacent
In a right-angled triangle, for a given non-right angled vertex, the adjacent side which is ''not'' the hypotenuse is referred to as '''the adjacent'''. In the above figure: : the '''adjacent''' to vertex $A$ is side $c$ : the '''adjacent''' to vertex $C$ is side $a$.
Definition:Geometric Figure/Plane Figure
A '''plane figure''' is a geometric figure embedded in the plane.
Definition:Geometric Figure/Three-Dimensional Figure
A '''three-dimensional figure''' is a geometric figure which cannot be embedded in the plane, but which '''can''' be embedded in three-dimensional space.
Definition:Geometric Figure/Diameter
The '''diameter''' of a geometric figure is the greatest length that can be formed between two opposite parallel straight lines that can be drawn tangent to its boundary.
Definition:Triangle (Geometry)/Isosceles/Base
The base of an isosceles triangle is specifically defined to be the side which is a different length from the other two. In the above diagram, $BC$ is the '''base'''.
Definition:Triangle (Geometry)/Isosceles/Base Angles
The two (equal) vertices adjacent to the base of an isosceles triangle are called the '''base angles'''. In the above diagram, $\angle ABC$ and $\angle ACB$ are the '''base angles'''.
Definition:Triangle (Geometry)/Isosceles/Apex
The vertex opposite the base of an isosceles triangle is called the '''apex''' of the triangle. In the above diagram, $A$ is the '''apex'''.
Definition:Triangle (Geometry)/Isosceles/Legs
The sides of an isosceles triangle which are adjacent to the apex are called the '''legs''' of the triangle. In the above diagram, $AB$ and $AC$ are the '''legs'''.
Definition:Triangle (Geometry)/Adjacent
The two sides of a triangle which form a particular vertex are referred to as '''adjacent''' to that angle. Similarly, the two vertices of a triangle to which a particular side contributes are referred to as '''adjacent''' to that side.
Definition:Triangle (Geometry)/Opposite
The side of a triangle which is ''not'' one of the sides adjacent to a particular vertex is referred to as its '''opposite'''. Thus, each vertex has an opposite side, and each side has an opposite vertex.
Definition:Triangle (Geometry)/Base
For a given triangle, one of the sides can be distinguished as being the '''base'''. It is immaterial which is so chosen. The usual practice is that the triangle is drawn so that the '''base''' is made horizontal, and at the bottom. In the above diagram, it would be conventional for the side $AC$ to be identified as the '''base'''.
Definition:Triangle (Geometry)/Apex
Having selected one side of a triangle to be the base, the opposite vertex to that base is called the '''apex'''. In the above diagram, if $AC$ is taken to be the base of $\triangle ABC$, then $B$ is the '''apex'''.
Definition:Triangle (Geometry)/Height
The '''height''' of a triangle is the length of a perpendicular from the apex to whichever side has been chosen as its base. That is, the length of the '''altitude''' so defined.
Definition:Triangle (Geometry)/Nomenclature
:300px The vertices of a triangle are conventionally labeled $A, B, C$ (or with other uppercase letters), and the sides with lowercase letters corresponding to the opposite vertex, as above. In order to emphasize that a particular vertex being referred to is in fact a vertex, the symbol $\angle$ is often placed by the letter corresponding to that vertex. Thus, for example: : $\angle A$ is adjacent to sides $b$ and $c$ : Side $a$ is adjacent to $\angle B$ and $\angle C$ : $\angle A$ is opposite side $a$ : Side $a$ is opposite $\angle A$.
Definition:Triangle (Geometry)/Equilateral
An '''equilateral triangle''' is a triangle in which all three sides are the same length: That is, a regular polygon with $3$ sides. :300px
Definition:Triangle (Geometry)/Scalene
A '''scalene triangle''' is a triangle in which all three sides are of different lengths. :500px
Definition:Triangle (Geometry)/Acute
An '''acute triangle''' is a triangle in which all three of the vertices are acute angles.
Definition:Triangle (Geometry)/Obtuse
An '''obtuse triangle''' is a triangle in which one of the vertices is an obtuse angle.
Definition:Quadrilateral/Trapezoid/Height
The '''height''' of a '''trapezoid''' is defined as the length of a line perpendicular to the bases. In the above diagram, the '''heights''' of the given trapezoids are indicated by the letter $h$.
Definition:Icosahedron/Regular
A '''regular icosahedron''' is an icosahedron whose $20$ faces are all congruent equilateral triangles.
Definition:Dodecahedron/Regular
A '''regular dodecahedron''' is a dodecahedron whose $12$ faces are all congruent regular pentagons.
Definition:Tetrahedron/Regular
A '''regular tetrahedron''' is a tetrahedron whose $4$ faces are all congruent equilateral triangles.
Definition:Plane Number
A '''plane number''' is the product of two (natural) numbers.  :''{{Definition:Euclid's Definitions - Book VII/16 - Plane Number}}'' {{EuclidDefRefNocat|VII|16|Plane Number}}
Definition:Solid Number
A '''solid number''' is the product of three (natural) numbers.  :''{{Definition:Euclid's Definitions - Book VII/17 - Solid Number}}'' {{EuclidDefRefNocat|VII|17|Solid Number}}
Definition:Incommensurable
Let $a, b \in \R_{>0}$ be (strictly) positive real numbers. $a$ and $b$ are '''incommensurable'''  $\dfrac a b$ is irrational.  :''{{Definition:Euclid's Definitions - Book X/1 - Commensurable}}'' {{EuclidDefRefNocat|X|1|Commensurable}}
Definition:Commensurable
Let $a, b \in \R_{>0}$ be (strictly) positive real numbers. $a$ and $b$ are '''commensurable''' iff $\dfrac a b$ is rational.  :''{{Definition:Euclid's Definitions - Book X/1 - Commensurable}}'' {{EuclidDefRefNocat|X|1|Commensurable}}
Definition:Inclination
=== Straight Line to Plane === {{Definition:Inclination/Straight Line to Plane}} === Plane to Plane === {{Definition:Inclination/Plane to Plane}}
Definition:Similar Inclination
{{EuclidDefinition|book=XI|def=7|name=Similarly Inclined}}
Definition:Sphere/Geometry/Axis
By definition, a sphere is made by turning a semicircle around a straight line. That straight line is called the '''axis of the sphere'''.  :''{{Definition:Euclid's Definitions - Book XI/15 - Axis of Sphere}}'' {{EuclidDefRefNocat|XI|15|Axis of Sphere}}
Definition:Sphere/Geometry/Diameter
The '''diameter of a sphere''' is the length of any straight line drawn from a point on the surface to another point on the surface through the center.
Definition:Cylinder/Axis
 :''{{Definition:Euclid's Definitions - Book XI/22 - Axis of Cylinder}}'' {{EuclidDefRefNocat|XI|22|Axis of Cylinder}} In the above diagram, the '''axis''' of the cylinder $ACBEFD$ is the straight line $GH$.
Definition:Cylinder/Base
 :''{{Definition:Euclid's Definitions - Book XI/23 - Base of Cylinder}}'' {{EuclidDefRefNocat|XI|23|Base of Cylinder}} In the above diagram, the '''bases''' of the cylinder $ACBEDF$ are the faces $ABC$ and $DEF$.
Definition:Cylinder/Similar Cylinders
Let $h_1$ and $h_2$ be the heights of two cylinders. Let $d_1$ and $d_2$ be the diameters of the bases of the two cylinders. Then the two cylinders are '''similar''' : :$\dfrac {h_1} {h_2} = \dfrac {d_1} {d_2}$  :''{{Definition:Euclid's Definitions - Book XI/24 - Similar Cones and Cylinders}}'' {{EuclidDefRefNocat|XI|24|Similar Cones and Cylinders}}
Definition:Segment of Circle/Base
The '''base''' of a segment of a circle is the straight line forming one of the boundaries of the seqment. In the above diagram, $AB$ is the '''base''' of the highlighted segment.
Definition:Segment of Circle/Angle of Segment
 :''{{:Definition:Euclid's Definitions - Book III/7 - Angle of Segment}}'' {{EuclidDefRefNocat|III|7|Angle of Segment}} That is, it is the angle the base makes with the circumference where they meet. It can also be defined as the angle between the base and the tangent to the circle at the end of the base: :300px
Definition:Sector/Angle
The '''angle of a sector''' is the angle between the two radii which delimit the sector. In the above diagram, the '''angle of the sector''' $BAC$ is $\theta$. The conjugate angle of $\theta$ also forms a '''sector''', denoted $CAB$ (see above).
Definition:Inclination/Straight Line to Plane
{{EuclidDefinition|book = XI|def = 5|name = Inclination of Straight Line}}
Definition:Inclination/Plane to Plane
{{EuclidDefinition|book = XI|def = 6|name = Inclination of Plane}}
Definition:Pyramid
:400px Let $P$ be a polygon. Let $Q$ be a point not in the plane of $P$. From each vertex of $P$, let lines be drawn to $Q$. The polyhedron which is contained by $P$ and the triangles formed by the sides of $P$ and the lines to $Q$ is a '''pyramid'''.
Definition:Right Circular Cone/Axis
Let $K$ be a right circular cone. Let point $A$ be the apex of $K$. Let point $O$ be the center of the base of $K$. Then the line $AO$ is the '''axis''' of $K$.  :''{{Definition:Euclid's Definitions - Book XI/19 - Axis of Cone}}'' {{EuclidDefRefNocat|XI|19|Axis of Cone}}
Definition:Right Circular Cone/Base
Let $\triangle AOB$ be a right-angled triangle such that $\angle AOB$ is the right angle. Let $K$ be the right circular cone formed by the rotation of $\triangle AOB$ around $OB$. Let $BC$ be the circle described by $B$. The '''base''' of $K$ is the plane surface enclosed by the circle $BC$.  :''{{Definition:Euclid's Definitions - Book XI/20 - Base of Cone}}'' {{EuclidDefRefNocat|XI|20|Base of Cone}}
Definition:Right Circular Cone/Acute-Angled
Let $K$ be a right circular cone. Then $K$ is '''acute-angled'''  the opening angle of $K$ is an acute angle.
Definition:Right Circular Cone/Right-Angled
Let $K$ be a right circular cone. Then $K$ is '''right-angled'''  the opening angle of $K$ is a right angle.
Definition:Right Circular Cone/Obtuse-Angled
Let $K$ be a right circular cone. Then $K$ is '''obtuse-angled'''  the opening angle of $K$ is an obtuse angle.
Definition:Circle/Center
 :''{{:Definition:Euclid's Definitions - Book I/16 - Center of Circle}}'' {{EuclidDefRefNocat|I|16|Center of Circle}} In the above diagram, the '''center''' is the point $A$.
Definition:Right Circular Cone/Similar Cones
Let $h_1$ and $h_2$ be the lengths of the axes of two right circular cones. Let $d_1$ and $d_2$ be the lengths of the diameters of the bases of the two right circular cones. Then the two right circular cones are '''similar''' : :$\dfrac {h_1} {h_2} = \dfrac {d_1} {d_2}$  :''{{Definition:Euclid's Definitions - Book XI/24 - Similar Cones and Cylinders}}'' {{EuclidDefRefNocat|XI|24|Similar Cones and Cylinders}}
Definition:Commensurable in Square
Let $a, b \in \R_{>0}$ be (strictly) positive real numbers. Then $a$ and $b$ are '''commensurable in square''' iff $\left ({\dfrac a b}\right)^2$ is rational.  :''{{Definition:Euclid's Definitions - Book X/2 - Commensurable in Square}}'' {{EuclidDefRefNocat|X|2|Commensurable in Square}}
Definition:Incommensurable in Square
Let $a, b \in \R_{>0}$ be (strictly) positive real numbers. Then $a$ and $b$ are '''incommensurable in square''' iff $\left ({\dfrac a b}\right)^2$ is irrational.  :''{{Definition:Euclid's Definitions - Book X/2 - Commensurable in Square}}'' {{EuclidDefRefNocat|X|2|Commensurable in Square}}
Definition:Binomial (Euclidean)
Let $a$ and $b$ be two (strictly) positive real numbers such that: : $(1): \quad \dfrac a b \notin \Q$ : $(2): \quad \left({\dfrac a b}\right)^2 \in \Q$ where $\Q$ denotes the set of rational numbers. Then $a + b$ is a '''binomial'''. {{:Euclid:Proposition/X/36}} === First Binomial === {{Definition:Binomial (Euclidean)/First Binomial}} === Second Binomial === {{Definition:Binomial (Euclidean)/Second Binomial}} === Third Binomial === {{Definition:Binomial (Euclidean)/Third Binomial}} === Fourth Binomial === {{Definition:Binomial (Euclidean)/Fourth Binomial}} === Fifth Binomial === {{Definition:Binomial (Euclidean)/Fifth Binomial}} === Sixth Binomial === {{Definition:Binomial (Euclidean)/Sixth Binomial}}
Definition:Binomial (Euclidean)/First Binomial
Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''first binomial''' : : $(1): \quad a \in \Q$ : $(2): \quad \dfrac {\sqrt {a^2 - b^2} } a \in \Q$ where $\Q$ denotes the set of rational numbers.  :''{{Definition:Euclid's Definitions - Book X (II)/1 - First Binomial}}'' {{EuclidDefRefNocat|X (II)|1|First Binomial}}
Definition:Circle/Circumference
The '''circumference''' of a circle is the line that forms its boundary.
Definition:Binomial (Euclidean)/Second Binomial
Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''second binomial''' : : $(1): \quad b \in \Q$ : $(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$ where $\Q$ denotes the set of rational numbers.  :''{{Definition:Euclid's Definitions - Book X (II)/2 - Second Binomial}}'' {{EuclidDefRefNocat|X (II)|2|Second Binomial}}
Definition:Binomial (Euclidean)/Third Binomial
Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''third binomial''' : : $(1): \quad a \notin \Q$ : $(2): \quad b \notin \Q$ : $(3): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$ where $\Q$ denotes the set of rational numbers.  :''{{Definition:Euclid's Definitions - Book X (II)/3 - Third Binomial}}'' {{EuclidDefRefNocat|X (II)|3|Third Binomial}}
Definition:Binomial (Euclidean)/Fourth Binomial
Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''fourth binomial''' : : $(1): \quad a \in \Q$ : $(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$ where $\Q$ denotes the set of rational numbers.  :''{{Definition:Euclid's Definitions - Book X (II)/4 - Fourth Binomial}}'' {{EuclidDefRefNocat|X (II)|4|Fourth Binomial}}
Definition:Binomial (Euclidean)/Fifth Binomial
Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''fifth binomial''' : : $(1): \quad b \in \Q$ : $(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$ where $\Q$ denotes the set of rational numbers.  :''{{Definition:Euclid's Definitions - Book X (II)/5 - Fifth Binomial}}'' {{EuclidDefRefNocat|X (II)|5|Fifth Binomial}}
Definition:Binomial (Euclidean)/Sixth Binomial
Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''sixth binomial''' : : $(1): \quad: a \notin \Q$ : $(2): \quad: b \notin \Q$ : $(3): \quad: \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$ where $\Q$ denotes the set of rational numbers.  :''{{Definition:Euclid's Definitions - Book X (II)/6 - Sixth Binomial}}'' {{EuclidDefRefNocat|X (II)|6|Sixth Binomial}}
Definition:Circle/Diameter
 :''{{:Definition:Euclid's Definitions - Book I/17 - Diameter of Circle}}'' {{EuclidDefRefNocat|I|17|Diameter of Circle}} In the above diagram, the line $CD$ is a '''diameter'''.
Definition:Circle/Radius
A '''radius''' of a circle is a straight line segment whose endpoints are the center and the circumference of the circle. In the above diagram, the line $AB$ is a '''radius'''.
Definition:Circle/Arc
An '''arc''' of a circle is any part of its circumference.
Definition:Circle/Semicircle
{{EuclidDefinition|book=I|def=18|name=Semicircle}}
Definition:Circle/Chord
A '''chord''' of a circle is a straight line segment whose endpoints are on the circumference of the circle. In the diagram above, the lines $CD$ and $EF$ are both '''chords'''.
Definition:Circle/Equality
 :''{{:Definition:Euclid's Definitions - Book III/1 - Equal Circles}}'' {{EuclidDefRefNocat|III|1|Equal Circles}}
Definition:Line/Endpoint
Each of the points at either end of a line segment is called an '''endpoint''' of that line segment. Similarly, the point at which an infinite half-line terminates is called '''the endpoint''' of that line.  :''{{:Definition:Euclid's Definitions - Book I/3 - Line Extremities}}'' {{EuclidDefRefNocat|I|3|Line Extremities}}
Definition:Apotome
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a > b$. Then $a - b$ is an '''apotome''' : :$(1): \quad \dfrac a b \notin \Q$ :$(2): \quad \paren {\dfrac a b}^2 \in \Q$ where $\Q$ denotes the set of rational numbers. {{:Euclid:Proposition/X/73}}
Definition:Apotome/First Apotome
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''first apotome''' : :$(1): \quad a \in \Q$ :$(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$ where $\Q$ denotes the set of rational numbers.  :''{{Definition:Euclid's Definitions - Book X (III)/1 - First Apotome}}'' {{EuclidDefRefNocat|X (III)|1|First Apotome}}
Definition:Apotome/Second Apotome
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''second apotome''' : :$(1): \quad b \in \Q$ :$(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$ where $\Q$ denotes the set of rational numbers.  :''{{Definition:Euclid's Definitions - Book X (III)/2 - Second Apotome}}'' {{EuclidDefRefNocat|X (III)|2|Second Apotome}}
Definition:Apotome/Third Apotome
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''third apotome''' : :$(1): \quad a \notin \Q$ :$(2): \quad b \notin \Q$ :$(3): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$ where $\Q$ denotes the set of rational numbers.  :''{{Definition:Euclid's Definitions - Book X (III)/3 - Third Apotome}}'' {{EuclidDefRefNocat|X (III)|3|Third Apotome}}
Definition:Line/Straight Line
 :''{{Definition:Euclid's Definitions - Book I/4 - Straight Line}}'' {{EuclidDefRefNocat|I|4|Straight Line}}
Definition:Line/Curve
A '''curve''' is a line which may or may not be straight.
Definition:Curve/Arc
An '''arc''' is a specific part of a curve.
Definition:Apotome/Fourth Apotome
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''fourth apotome''' : :$(1): \quad a \in \Q$ :$(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$ where $\Q$ denotes the set of rational numbers.  :''{{Definition:Euclid's Definitions - Book X (III)/4 - Fourth Apotome}}'' {{EuclidDefRefNocat|X (III)|4|Fourth Apotome}}