Symbols for Basic Propositions
Used in the First Four Books of Apollonius' Conics
In this section, propositions and definitions
(mostly those of Euclid's Elements)
used in the first four Books of the Conics are listed with short explanations.
The propositions from Euclid's Data come first,
followed by those from the Elements.
Legenda
Data.3
Euclid's Data, proposition 3
E.6-1
Euclid's Elements, Book 6, proposition 1
Hypothesis falsa
indicates beginning of reductio ad absurdum
Propositions number may be followed by one or more
of the following:
Dir and Conv (conv)
Proposition 6-16 consists of an enunciation (6-16-Dir)
and its converse (6-16-Conv). The converse of 3-35 is
not stated in the Elements but is used by Pappus;
this is referred to as 3-35-conv (note the difference
of Conv and conv).
A, B, C
Proposition 5-7 consists of two enunciations which are
not converses reciprocally. They are referred to as
5-7-A, 5-7-B
Cor
Corollaries are indicated by Cor.
x, x2, lemma, cor, ..., x-conv,
For extensions of a proposition, x, x2, ...
If the extension is concerned with inequality while the
proposition itself deals with equality, ``ineq'' is
used insted of ``x''.
Some other extentions are classified as lemma and
cor (corollary; note that ``Cor'' indicates corollary explicitely
stated in the text. Therefore, 3-35-x, 5-17-ineq, 6-23-lemma, etc.
def
Of course def (definition) means definitions such as E.1-def-10.
As Apollonius seems to assume the definition of concaveness and convexness
in some of the propositions in Book 4, these assumptions are
indicated as ``Definition of concaveness''
For compounded ratios, which are used in E.6-23 with no
genuine definition, their definition is reconstructed and
referred to as 6-23-def.
The following propositions are so particular that they could not
be related to a single proposition of Euclid:
2-B, 5-x-1, 5-x-2
List of Propositions and Definitions
Definition of concaveness (chord)
Definition of concaveness (tangent)
Implicit definition of concaveness.
Hypothesis falsa
indicates the beginning of the
proof by reductio ad absurdum.
Data.1
If a and b are given, then a:b is given.
Data.3
If a and b are given, then a+b is given
Data.8
If a:b and c:b are given, then a:c is given
Data.25
If two lines (not necessarily straight lines) are given in position,
then the point of intersection is given.
Data.26
If two point A and B are given in position, then the straight line AB
is given in position and magnitude.
Data.27
If a straight line is given in position and magnitude, and one of its
ends is given, then the other end will also be given.
Data.27-x
If a line which is given in magnitude and in position, is divided into a given ratio
at a point, then the point will be given.
Data.27-x2
(Same as 27-x; the point is in the extension of the given line)
0
Data.28 (cf. E.1-31)
If, through a given point, a parallel straight line is drawn to
a straight line given in position, then the line thus drawn is given
in position.
Data.29
If, to a straight line given in position, and at a given point
in it, a line is drawn making a given angle, then the line thus
drawn is given in position.
Data.30
If, from a given point, to a straight line given in position,
a straight line is drawn making a given angle, then the line thus drawn
is given in position.
Data.40 (cf. E.6--4)
If each of the angles of a triangle is given in magnitude,
then the triangle is given in form.
Data.41 (cf. E.6--6)
If a triangle has one given angle, and the sides about the
given angle have given ratio to each other, then the triangle
is given in form.
Data.50 (cf. E.6--22)
If two straight lines have given ratio to each other, then
the rectilineal figures similar and similarly described upon
them will have the given ratio to each other.
Data.57
If given area is applied to a given straight line in given angle,
then the width of application is given.
E.1-def-10, 11, 12, 15, 17, 19, 23, 33
E.1-post.1-5
E.1-3, 4, 6, 8, 10, 11, 12
E.1-13
E.1-14-x
= 1-13 + 1-14 = 1-15-conv
E.1-15, 16, 17, 18, 20, 21, 22, 23, 26
E.1-27
E.1-27-x
If two straight lines are not parallel, and a straight line falling
on them does NOT make the alternate angles equal to each other, nor
the exterior angle equal to the interior and opposite angle.
E.1-28
E.1-29
For E.1-29 + E.6--4, only E.6--4 is noted.
E.1-29-x
If a straight line meets one of parallel straight lines,
it will also meet others.
E.1-30, 31, 32, 33
E.1-34
E.1-34-def
Definition of parallelogram.
E.1-34-x
Diameters of a parallelogram bisect each other.
E.1-34-x2
If from two parallel straight lines, two straight
lines cut off unequal segments, then the straight lines
will meet, when produced, on the side of the shorter segment.
E.1-36, 38, 39
E.1-41
E.1-41-trap
Area of a trapezium.
E.1-43
E.1-44-x
Parabolic application of an area equal to a given rectilinear
figure (not necessarily triangle)
E.1-44-x-ineq
Apply an area greater than a given figure.
E.1-47
E.2-3
E.2-5
E.2-5-conv
E.2-5-cor
If a straight lines is divided into two segments
by a point on it, then the nearer the dividing point
to the midpoint, the greater the rectangle contained
by the segments.
E.2-6
E.2-6-conv
E.2-6-cor
(2-5-cor for the point on the extension of the straight line)
E.2-9, 10
E.2-12-cor
In obtuse-angled triangles, the square on the side
subtending the obtuse angle is greater than the
squares on the sicdes ontaining the obtuse angle.
E.2-14
E.2-B
Complicated "geometric algebra"
E.3-1, 1-Cor, 2, 3, 15, 16, 21, 27, 30, 31, 33
E.3-35
E.3-36-x
If from a point outside a circle, two straigh lines
fall on the circle each cutting the circle,
then the rectangles contained by the whole line
and the segment between the point and the convex circumference
are equal to each other.
E.5-def-5
E.5-4
E.5-4-ineq
If a:b>c:d, then ma:b>mc:d, etc.
E.5-7-A
If a=b then a:c::b:c.\\
Substitution of a term in a proportion, e.g.,
a:b::c:d\quad\mbox{and}{\quad}e=a \Longrightarrow e:b::c:d
is NOT counted, though 5-7-A and 5-11 are implicitly used.
5-7-B
If a=b then c:a::c:bD(The same criterion as 5-7-A applies).
5-7-Cor
If magnitudes are proportional, they will also be proportional
conversely.
5-7-cor-1
(Mueller VD) If a=b and c=d, then a:b=c:d
5-7-cor-1m
ma:a::mb:b.
E.5-7-cor-2
(=5-7-cor-1 + 5-16)\\
If a=A, b=B, then a:b::A:B
5-8-A
If a>c then a:b>c:b.
5-8-B
If a>c then b:c>b:a.
5-9-A
If a:b::c:b then a=c.
5-9-B
If a:b::a:c then b=c
5-10-A
If a:b>c:b then a>c.
5-10-B
If a:b>a:c then bE.5-11
E.5-11-x
(cf.\ E.5-13-x)
If a:b::c:d, then any term can be replaced by anything equal
to it; e.g., A:b::c:d if a=A.
E.5-12
E.5-12-x
If a:A=b:B=c:C and a+b=c, then A+B=C, etc.
E.5-13
E.5-13-x
(cf.\ E.5-11-x)
If a:b>c:d, then any term can be replaced by anything equal
to it; e.g., A:b>c:d if a=A.
E.5-14
If a:b::c:d, then according to a>==E.5-14-a
(See [Gardies, 1991] and [Saito, 1994])\\
If a:b::c:d, then according to a>==E.5-14-am
(cf. E.5-7-cor-1m)\\
If a:b::c:d and a=mb, then c=md
E.5-15
E.5-16, 5-16-ineq, 5-17, 5-17-ineq, 5-18
E.5-19, 5-19-Cor, 5-19-Cor-ineq
E.5-22
E.5-x-1
(cf. Data.27-x)\\
If two points are taken on a straight line, they will divide the
line in different ratios.
E.5-x-2
If A, B, C, D are on a straight line in this order, and if
AB:BC::AD:DC, then no other point than C between BD satisfies
this proportion.
E.6-def.1
E.6-1
E.6-2-Dir
(Figure 2) If k//l, then a:b::c:d.
E.6-2-Dir-eq
(Figure 2) If k//l and a=b, then c=d.
E.6-2-Conv
(Figure 2) If a:b::c:d, then k//l.
E.6-2-Conv-eq
(Figure 2) If a=b and c=d, then k//l
E.6-2-x
(Figure 3, k//l) a:b::c:d.
E.6-2-x-eq
(Figure 3, k//l) If a=b, then c=d.
E.6-2-x2
(Figure 4, k//l//m) a:b::c:d
E.6-2-x2-eq
(Figure 4, k//l//m) If a=b, then c=d.
E.6-4
(see also E.1-29)
E.6-4-ineq
E.6-6
E.6-6-ineq
E.6-6-x, E.6-6-x2
See Pappus 7-257 and 7-255 respectively.
E.6-8-coroll
E.6-8-x
(Figure 5)
If from the right angle A of a rectangular triangle
ABC a perpendicular AD is drawn, then:
sq(\mbox{AD})=rec(\mbox{BD},\mbox{DC}),
sq(\mbox{AB})=rec(\mbox{CB},\mbox{BD}),
sq(\mbox{AC})=rec(\mbox{BC},\mbox{CD}).
E.6-8-x2
(Figure 5; cf.\ Pappus 7-237)\\
If from any point A on a curve perpendicular AD is drawn to a straight line
BC, and if the square on AD is always equal to the
rectangle contained by BDC, then the curve
is (semi)circle.
E.6-9
E.6-10
E.6-10-x
Given a straight line AB, find a point C in the extension of AB
such that AC:CB is given.
E.6-12
(= Pappus 7-44)
E.6-12-x
To find a line "d" such that a:b::sq(c):sq(d) etc.
E.6-13
E.6-14-cor
(cf. Eutocius in Conica 1-49)\\
Equiangular parallelograms have to one another
the ratio of rectangles having the same sides as the parallelograms.
E.6-15-Dir, E.6-15-Conv-x
E.6-16-Dir, E.6-16-Dir-ineq
E.6-16-Conv, E.6-16-Conv-ineq
E.6-17-Dir, E.6-17-Conv
E.6-19-Cor, E.6-19-Cor-conv
E.6-22-Dir, E.6-22-Dir-ineq
E.6-22-Conv
E.6-22-x
(= E.6-22-Dir + E.5-16)
E.6-23
E.6-23-def
(a:b)(b:c)::a:c. Definition of compounded ratio.
E.6-23-lemma
Uniqueness of compounded ratio:
if a:b::x:y and c:d::z:w, then (a:b)(c:d)::(x:y)(z:w)
E.6-23-lemma-2
If (a:b)(c:d)::(x:y)(z:w), and a:b::x:y, then c:d::z:w.