% definitions.tex --- Book I definitions (the 23). Each is a `scope`
% block: definitions delimit terms used by later claims and edges, but
% they're not themselves assertions about the world.

\section*{Book I: Definitions}
\label{sec:def-bookI}

\begin{scope}[Definition I.1: point]
\label{def:I.1}
A point is that which has no part.
\end{scope}

\begin{scope}[Definition I.2: line]
\label{def:I.2}
A line is breadthless length.
\end{scope}

\begin{scope}[Definition I.3: extremities of a line]
\label{def:I.3}
The extremities of a line are points.
\end{scope}

\begin{scope}[Definition I.4: straight line]
\label{def:I.4}
A straight line is a line which lies evenly with the points on itself.
\end{scope}

\begin{scope}[Definition I.5: surface]
\label{def:I.5}
A surface is that which has length and breadth only.
\end{scope}

\begin{scope}[Definition I.6: extremities of a surface]
\label{def:I.6}
The extremities of a surface are lines.
\end{scope}

\begin{scope}[Definition I.7: plane surface]
\label{def:I.7}
A plane surface is a surface which lies evenly with the straight lines
on itself.
\end{scope}

\begin{scope}[Definition I.8: plane angle]
\label{def:I.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.
\end{scope}

\begin{scope}[Definition I.9: rectilineal angle]
\label{def:I.9}
And when the lines containing the angle are straight, the angle is
called rectilineal.
\end{scope}

\begin{scope}[Definition I.10: right angle]
\label{def:I.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 called a perpendicular to
that on which it stands.
\end{scope}

\begin{scope}[Definition I.11: obtuse angle]
\label{def:I.11}
An obtuse angle is an angle greater than a right angle.
\end{scope}

\begin{scope}[Definition I.12: acute angle]
\label{def:I.12}
An acute angle is an angle less than a right angle.
\end{scope}

\begin{scope}[Definition I.13: boundary]
\label{def:I.13}
A boundary is that which is an extremity of anything.
\end{scope}

\begin{scope}[Definition I.14: figure]
\label{def:I.14}
A figure is that which is contained by any boundary or boundaries.
\end{scope}

\begin{scope}[Definition I.15: circle]
\label{def:I.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.
\end{scope}

\begin{scope}[Definition I.16: centre]
\label{def:I.16}
And the point is called the centre of the circle.
\end{scope}

\begin{scope}[Definition I.17: diameter]
\label{def:I.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 the circle.
\end{scope}

\begin{scope}[Definition I.18: semicircle]
\label{def:I.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.
\end{scope}

\begin{scope}[Definition I.19: rectilineal figures]
\label{def:I.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 those contained by more than four
straight lines.
\end{scope}

\begin{scope}[Definition I.20: kinds of trilateral]
\label{def:I.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 that which has its three
sides unequal.
\end{scope}

\begin{scope}[Definition I.21: kinds of trilateral by angle]
\label{def:I.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 that which has its three angles
acute.
\end{scope}

\begin{scope}[Definition I.22: kinds of quadrilateral]
\label{def:I.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 equilateral but not right-angled;
and a rhomboid that which has its opposite sides and angles equal to
one another but is neither equilateral nor right-angled. And let
quadrilaterals other than these be called trapezia.
\end{scope}

\begin{scope}[Definition I.23: parallel straight lines]
\label{def:I.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.
\end{scope}

% ===== Book II definitions =====

\begin{scope}[Definition II.1: rectangle]
\label{def:II.1}
Any rectangular parallelogram is said to be contained by the two
straight lines containing the right angle.
\end{scope}

\begin{scope}[Definition II.2: gnomon]
\label{def:II.2}
And in any parallelogrammic area let any one whatever of the
parallelograms about its diameter, with the two complements, be
called a gnomon.
\end{scope}

% ===== Book III definitions =====

\begin{scope}[Definition III.1: equal circles]
\label{def:III.1}
Equal circles are those whose diameters are equal, or whose radii are
equal.
\end{scope}

\begin{scope}[Definition III.2: tangent line]
\label{def:III.2}
A straight line is said to touch a circle which, meeting the circle
and being produced, does not cut the circle.
\end{scope}

\begin{scope}[Definition III.3: tangent circles]
\label{def:III.3}
Circles are said to touch one another which, meeting one another, do
not cut one another.
\end{scope}

\begin{scope}[Definition III.4: equidistant chords]
\label{def:III.4}
In a circle, straight lines are said to be equally distant from the
centre when the perpendiculars drawn to them from the centre are
equal.
\end{scope}

\begin{scope}[Definition III.5: more distant chord]
\label{def:III.5}
And that straight line is said to be at a greater distance on which
the greater perpendicular falls.
\end{scope}

\begin{scope}[Definition III.6: segment of a circle]
\label{def:III.6}
A segment of a circle is the figure contained by a straight line and
a circumference of a circle.
\end{scope}

\begin{scope}[Definition III.7: angle of a segment]
\label{def:III.7}
An angle of a segment is that contained by a straight line and a
circumference of a circle.
\end{scope}

\begin{scope}[Definition III.8: angle in a segment]
\label{def:III.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 which is the base of the
segment, is contained by the straight lines so joined.
\end{scope}

\begin{scope}[Definition III.9: standing on an arc]
\label{def:III.9}
And, when the straight lines containing the angle cut off an arc, the
angle is said to stand upon that arc.
\end{scope}

\begin{scope}[Definition III.10: sector]
\label{def:III.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 arc cut off by them.
\end{scope}

\begin{scope}[Definition III.11: similar segments]
\label{def:III.11}
Similar segments of circles are those which admit equal angles, or in
which the angles are equal to one another.
\end{scope}

% ===== Book V definitions (Eudoxean proportion) =====

\begin{scope}[Definition V.1: part]
\label{def:V.1}
A magnitude is a part of a magnitude, the less of the greater, when
it measures the greater.
\end{scope}

\begin{scope}[Definition V.2: multiple]
\label{def:V.2}
The greater is a multiple of the less when it is measured by the less.
\end{scope}

\begin{scope}[Definition V.3: ratio]
\label{def:V.3}
A ratio is a sort of relation in respect of size between two
magnitudes of the same kind.
\end{scope}

\begin{scope}[Definition V.4: having a ratio]
\label{def:V.4}
Magnitudes are said to have a ratio to one another which are capable,
when multiplied, of exceeding one another (the Archimedean property).
\end{scope}

\begin{scope}[Definition V.5: same ratio (Eudoxean equality of ratios)]
\label{def:V.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 equimultiples whatever of the
second and fourth, the former equimultiples alike exceed, are alike
equal to, or alike fall short of, the latter equimultiples respectively
taken in corresponding order.
\end{scope}

\begin{scope}[Definition V.6: proportional]
\label{def:V.6}
Let magnitudes which have the same ratio be called proportional.
\end{scope}

\begin{scope}[Definition V.7: greater ratio]
\label{def:V.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 the first is said
to have a greater ratio to the second than the third has to the fourth.
\end{scope}

\begin{scope}[Definition V.8: proportion (three terms)]
\label{def:V.8}
A proportion in three terms is the least possible.
\end{scope}

\begin{scope}[Definition V.9: duplicate ratio]
\label{def:V.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.
\end{scope}

\begin{scope}[Definition V.10: triplicate ratio]
\label{def:V.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, in continual proportion of any number of
magnitudes.
\end{scope}

\begin{scope}[Definition V.11: corresponding magnitudes]
\label{def:V.11}
Antecedents are said to correspond to antecedents, and consequents to
consequents.
\end{scope}

\begin{scope}[Definition V.12: alternate ratio]
\label{def:V.12}
Alternate ratio means taking the antecedent in relation to the
antecedent and the consequent in relation to the consequent.
\end{scope}

\begin{scope}[Definition V.13: inverse ratio]
\label{def:V.13}
Inverse ratio means taking the consequent as antecedent in relation
to the antecedent as consequent.
\end{scope}

\begin{scope}[Definition V.14: composition of a ratio]
\label{def:V.14}
Composition of a ratio means taking the antecedent together with the
consequent as one in relation to the consequent by itself.
\end{scope}

\begin{scope}[Definition V.15: separation of a ratio]
\label{def:V.15}
Separation of a ratio means taking the excess by which the antecedent
exceeds the consequent in relation to the consequent by itself.
\end{scope}

\begin{scope}[Definition V.16: conversion of a ratio]
\label{def:V.16}
Conversion of a ratio means taking the antecedent in relation to the
excess by which the antecedent exceeds the consequent.
\end{scope}

\begin{scope}[Definition V.17: ratio ex aequali]
\label{def:V.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 the last of the first
magnitudes, so is the first to the last of the second magnitudes.
\end{scope}

\begin{scope}[Definition V.18: perturbed proportion]
\label{def:V.18}
A perturbed proportion arises when, there being three magnitudes and
another set equal to them in multitude, as antecedent is to consequent
among the first magnitudes, so is antecedent to consequent among the
second magnitudes, while as the consequent is to a third among the
first magnitudes, so is a third to the antecedent among the second
magnitudes.
\end{scope}

% ===== Book VII definitions (Number theory) =====

\begin{scope}[Definition VII.1: unit]
\label{def:VII.1}
A unit is that by virtue of which each of the things that exist is
called one.
\end{scope}

\begin{scope}[Definition VII.2: number]
\label{def:VII.2}
A number is a multitude composed of units.
\end{scope}

\begin{scope}[Definition VII.3: part of a number]
\label{def:VII.3}
A number is a part of a number, the less of the greater, when it
measures the greater.
\end{scope}

\begin{scope}[Definition VII.4: parts]
\label{def:VII.4}
But parts when it does not measure it.
\end{scope}

\begin{scope}[Definition VII.5: multiple]
\label{def:VII.5}
The greater number is a multiple of the less when it is measured by
the less.
\end{scope}

\begin{scope}[Definition VII.6: even number]
\label{def:VII.6}
An even number is that which is divisible into two equal parts.
\end{scope}

\begin{scope}[Definition VII.7: odd number]
\label{def:VII.7}
An odd number is that which is not divisible into two equal parts, or
that which differs by a unit from an even number.
\end{scope}

\begin{scope}[Definition VII.8: even-times even]
\label{def:VII.8}
An even-times even number is that which is measured by an even number
according to an even number.
\end{scope}

\begin{scope}[Definition VII.9: even-times odd]
\label{def:VII.9}
An even-times odd number is that which is measured by an even number
according to an odd number.
\end{scope}

\begin{scope}[Definition VII.10: odd-times odd]
\label{def:VII.10}
An odd-times odd number is that which is measured by an odd number
according to an odd number.
\end{scope}

\begin{scope}[Definition VII.11: prime number]
\label{def:VII.11}
A prime number is that which is measured by a unit alone.
\end{scope}

\begin{scope}[Definition VII.12: relatively prime]
\label{def:VII.12}
Numbers prime to one another are those which are measured by a unit
alone as a common measure.
\end{scope}

\begin{scope}[Definition VII.13: composite number]
\label{def:VII.13}
A composite number is that which is measured by some number.
\end{scope}

\begin{scope}[Definition VII.14: numbers composite to one another]
\label{def:VII.14}
Numbers composite to one another are those which are measured by some
number as a common measure.
\end{scope}

\begin{scope}[Definition VII.15: multiply]
\label{def:VII.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.
\end{scope}

\begin{scope}[Definition VII.16: plane number]
\label{def:VII.16}
When two numbers having multiplied one another make some number, the
number so produced is called plane, and its sides are the numbers
which have multiplied one another.
\end{scope}

\begin{scope}[Definition VII.17: solid number]
\label{def:VII.17}
When three numbers having multiplied one another make some number,
the number so produced is solid, and its sides are the numbers which
have multiplied one another.
\end{scope}

\begin{scope}[Definition VII.18: square number]
\label{def:VII.18}
A square number is equal multiplied by equal, or a number which is
contained by two equal numbers.
\end{scope}

\begin{scope}[Definition VII.19: cube number]
\label{def:VII.19}
A cube number is equal multiplied by equal and again by equal, or a
number which is contained by three equal numbers.
\end{scope}

\begin{scope}[Definition VII.20: proportional numbers]
\label{def:VII.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.
\end{scope}

\begin{scope}[Definition VII.21: similar plane and solid numbers]
\label{def:VII.21}
Similar plane and solid numbers are those which have their sides
proportional.
\end{scope}

\begin{scope}[Definition VII.22: perfect number]
\label{def:VII.22}
A perfect number is that which is equal to the sum of its own parts
(its proper divisors).
\end{scope}

% ===== Book X definitions (group 1, before X.1) =====

\begin{scope}[Definition X.1: commensurable magnitudes]
\label{def:X.1}
Those magnitudes are said to be commensurable which are measured by
the same measure, and those incommensurable which cannot have any
common measure.
\end{scope}

\begin{scope}[Definition X.2: commensurable in square]
\label{def:X.2}
Straight lines are commensurable in square when the squares on them
are measured by the same area, and incommensurable in square when the
squares on them cannot possibly have any area as a common measure.
\end{scope}

\begin{scope}[Definition X.3: rational and irrational straight lines]
\label{def:X.3}
With these hypotheses, it is proved that there exist straight lines
infinite in multitude which are commensurable and incommensurable
respectively, some in length only, and others in square also, with
an assigned straight line.  Let the assigned straight line be called
rational, and those straight lines which are commensurable with it,
whether in length and in square or in square only, rational, but
those which are incommensurable with it irrational.
\end{scope}

\begin{scope}[Definition X.4: rational and irrational areas]
\label{def:X.4}
And let the square on the assigned straight line be called rational
and those areas which are commensurable with it rational, but those
which are incommensurable with it irrational, and the straight lines
which produce them irrational --- that is, in case the areas are
squares, the sides themselves; in other cases, the straight lines on
which the rectangles equal to the areas would be applied.
\end{scope}

% ===== Book X definitions (group 2, before X.48) =====

\begin{scope}[Definition X(2).1: binomial straight line]
\label{def:X.II.1}
Given a rational straight line and a binomial, divided into its terms,
let the square of the greater term be greater than the square of the
lesser by the square of a straight line commensurable in length with
the greater. Then if the greater term is commensurable in length with
the assigned rational straight line, the whole is called a first binomial.
\end{scope}

\begin{scope}[Definition X(2).2: second binomial]
\label{def:X.II.2}
If the lesser term is commensurable in length with the assigned
rational straight line, the whole is called a second binomial.
\end{scope}

\begin{scope}[Definition X(2).3: third binomial]
\label{def:X.II.3}
If neither term is commensurable in length with the assigned rational
straight line, the whole is called a third binomial.
\end{scope}

\begin{scope}[Definition X(2).4: fourth binomial]
\label{def:X.II.4}
If the square of the greater term exceeds the square of the lesser by
the square of a line incommensurable in length with the greater, and
the greater term is commensurable in length with the assigned
rational straight line, the whole is called a fourth binomial.
\end{scope}

\begin{scope}[Definition X(2).5: fifth binomial]
\label{def:X.II.5}
If, in the same case, the lesser term is commensurable in length with
the assigned rational straight line, the whole is called a fifth
binomial.
\end{scope}

\begin{scope}[Definition X(2).6: sixth binomial]
\label{def:X.II.6}
If neither term is commensurable in length with the assigned rational
straight line, the whole is called a sixth binomial.
\end{scope}

% ===== Book X definitions (group 3, before X.85) =====

\begin{scope}[Definition X(3).1: first apotome]
\label{def:X.III.1}
Given a rational straight line and an apotome (i.e. a difference of
two rationals commensurable in square only), if the square of the
whole is greater than the square of the annex by the square of a
straight line commensurable in length with the whole, and the whole
is commensurable in length with the assigned rational straight line,
the apotome is called a first apotome.
\end{scope}

\begin{scope}[Definition X(3).2: second apotome]
\label{def:X.III.2}
If the annex is commensurable in length with the assigned rational
straight line, the apotome is called a second apotome.
\end{scope}

\begin{scope}[Definition X(3).3: third apotome]
\label{def:X.III.3}
If neither the whole nor the annex is commensurable in length with
the assigned rational straight line, the apotome is called a third
apotome.
\end{scope}

\begin{scope}[Definition X(3).4: fourth apotome]
\label{def:X.III.4}
If the square of the whole exceeds the square of the annex by the
square of a straight line incommensurable in length with the whole,
and the whole is commensurable in length with the assigned rational
straight line, the apotome is called a fourth apotome.
\end{scope}

\begin{scope}[Definition X(3).5: fifth apotome]
\label{def:X.III.5}
If, in the same case, the annex is commensurable in length with the
assigned rational straight line, the apotome is called a fifth apotome.
\end{scope}

\begin{scope}[Definition X(3).6: sixth apotome]
\label{def:X.III.6}
If neither the whole nor the annex is commensurable in length with
the assigned rational straight line, the apotome is called a sixth
apotome.
\end{scope}

% ===== Book XI definitions (Solid geometry) =====

\begin{scope}[Definition XI.1: solid]
\label{def:XI.1}
A solid is that which has length, breadth, and depth.
\end{scope}

\begin{scope}[Definition XI.2: extremity of a solid]
\label{def:XI.2}
An extremity of a solid is a surface.
\end{scope}

\begin{scope}[Definition XI.3: line at right angles to a plane]
\label{def:XI.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.
\end{scope}

\begin{scope}[Definition XI.4: plane at right angles to a plane]
\label{def:XI.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 plane.
\end{scope}

\begin{scope}[Definition XI.5: inclination of a line to a plane]
\label{def:XI.5}
The inclination of a straight line to a plane is, assuming a
perpendicular drawn from the extremity of the straight line which is
elevated above the plane to the plane and a straight line joined from
the foot of the perpendicular to the extremity of the straight line
which is in the plane, the angle contained by the straight line so
drawn and the straight line standing up.
\end{scope}

\begin{scope}[Definition XI.6: inclination of plane to plane]
\label{def:XI.6}
The inclination of a plane to a plane is the acute angle contained
by the straight lines drawn at right angles to the common section at
the same point, one in each of the planes.
\end{scope}

\begin{scope}[Definition XI.7: similarly inclined planes]
\label{def:XI.7}
A plane is said to be similarly inclined to a plane as another to
another when the said angles of the inclinations are equal to one
another.
\end{scope}

\begin{scope}[Definition XI.8: parallel planes]
\label{def:XI.8}
Parallel planes are those which do not meet.
\end{scope}

\begin{scope}[Definition XI.9: similar solid figures]
\label{def:XI.9}
Similar solid figures are those contained by similar planes equal in
multitude.
\end{scope}

\begin{scope}[Definition XI.10: equal and similar solid figures]
\label{def:XI.10}
Equal and similar solid figures are those contained by similar planes
equal in multitude and in magnitude.
\end{scope}

\begin{scope}[Definition XI.11: solid angle]
\label{def:XI.11}
A solid angle is the inclination constituted by more than two lines
which meet one another and are not in the same surface, towards all
the lines.  Otherwise: a solid angle is that which is contained by
more than two plane angles which are not in the same plane and are
constructed to one point.
\end{scope}

\begin{scope}[Definition XI.12: pyramid]
\label{def:XI.12}
A pyramid is a solid figure contained by planes which is constructed
from one plane to one point.
\end{scope}

\begin{scope}[Definition XI.13: prism]
\label{def:XI.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.
\end{scope}

\begin{scope}[Definition XI.14: sphere]
\label{def:XI.14}
When a semicircle with fixed diameter is carried round and restored
again to the same position from which it began to be moved, the
figure so comprehended is a sphere.
\end{scope}

\begin{scope}[Definition XI.15: axis of a sphere]
\label{def:XI.15}
The axis of the sphere is the straight line which remains fixed and
about which the semicircle is turned.
\end{scope}

\begin{scope}[Definition XI.16: centre of a sphere]
\label{def:XI.16}
The centre of the sphere is the same as that of the semicircle.
\end{scope}

\begin{scope}[Definition XI.17: diameter of a sphere]
\label{def:XI.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.
\end{scope}

\begin{scope}[Definition XI.18: cone]
\label{def:XI.18}
When, one side of those about the right angle in a right-angled
triangle remaining fixed, the triangle is carried round and restored
again to the same position from which it began to be moved, the
figure so comprehended is a cone.  And if the straight line which
remains fixed is equal to the remaining side about the right angle
which is carried round, the cone will be right-angled; if less,
obtuse-angled; and if greater, acute-angled.
\end{scope}

\begin{scope}[Definition XI.19: axis of a cone]
\label{def:XI.19}
The axis of the cone is the straight line which remains fixed and
about which the triangle is turned.
\end{scope}

\begin{scope}[Definition XI.20: base of a cone]
\label{def:XI.20}
And the base is the circle described by the straight line which is
carried round.
\end{scope}

\begin{scope}[Definition XI.21: cylinder]
\label{def:XI.21}
When, one side of those about the right angle in a rectangular
parallelogram remaining fixed, the parallelogram is carried round and
restored again to the same position from which it began to be moved,
the figure so comprehended is a cylinder.
\end{scope}

\begin{scope}[Definition XI.22: axis of a cylinder]
\label{def:XI.22}
The axis of the cylinder is the straight line which remains fixed and
about which the parallelogram is turned.
\end{scope}

\begin{scope}[Definition XI.23: bases of a cylinder]
\label{def:XI.23}
The bases are the circles described by the two sides opposite to one
another which are carried round.
\end{scope}

\begin{scope}[Definition XI.24: similar cones and cylinders]
\label{def:XI.24}
Similar cones and cylinders are those in which the axes and the
diameters of the bases are proportional.
\end{scope}

\begin{scope}[Definition XI.25: cube]
\label{def:XI.25}
A cube is a solid figure contained by six equal squares.
\end{scope}

\begin{scope}[Definition XI.26: octahedron]
\label{def:XI.26}
An octahedron is a solid figure contained by eight equal and
equilateral triangles.
\end{scope}

\begin{scope}[Definition XI.27: icosahedron]
\label{def:XI.27}
An icosahedron is a solid figure contained by twenty equal and
equilateral triangles.
\end{scope}

\begin{scope}[Definition XI.28: dodecahedron]
\label{def:XI.28}
A dodecahedron is a solid figure contained by twelve equal,
equilateral, and equiangular pentagons.
\end{scope}

% ===== Book XIII definitions (5 supplementary defs) =====

\begin{scope}[Definition XIII.1: extreme and mean ratio]
\label{def:XIII.1}
A straight line is said to have been cut in extreme and mean ratio
when, as the whole line is to the greater segment, so is the greater
to the lesser.
\end{scope}

\begin{scope}[Definition XIII.2: height of a figure]
\label{def:XIII.2}
The height of any figure is the perpendicular drawn from the vertex
to the base.
\end{scope}

\begin{scope}[Definition XIII.3: medial straight line]
\label{def:XIII.3}
A medial straight line is the mean proportional between two rational
straight lines commensurable in square only.
\end{scope}

\begin{scope}[Definition XIII.4: minor straight line]
\label{def:XIII.4}
A minor straight line is the difference of two straight lines
incommensurable in square such that the sum of the squares on them
is rational, but the rectangle contained by them is medial.
\end{scope}

\begin{scope}[Definition XIII.5: composite irrational]
\label{def:XIII.5}
A straight line which produces with a rational area a medial whole
is the irrational straight line such that the square on it added to
a rational area makes the whole medial.
\end{scope}
