Index of the Propositions Used in Book 7 of Pappus' Collection
Ken SAITO

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Part 1-2: List of the Propositions used in Pappus Book 7.

5.1 Index Part 1: List of the propositions used in Pappus Book 7

• Elements

1-def-10, 1-def-11, 1-def-12, 1-def-15, 1-def-17

1-1, 1-2, 1-3, 1-4

1-4-x

If two rectangular triangles have two sides equal to two sides, they are congruent.

1-5, 1-6, 1-8, 1-9, 1-10, 1-11, 1-12, 1-13, 1-14

1-14-x

= 1-13 + 1-14 = 1-15-conv

1-15, 1-18, 1-19, 1-21, 1-23, 1-26, 1-27, 1-28, 1-29

1-29-x

If a straight line meets one of parallel straight lines, it will meet others.

1-30, 1-31,

1-32-A

The exterior angle of a triangle is equal to the two interior opposite angles.

1-32-B

The three interior angles of a triangles are equal to two right angles.

1-32-cor-1

The exterior angle of a triangle is greater than either of the interior opposite angles.

1-32-cor-2

In a rectangular triangle, the remaining two angles are acute.

1-32-cor-3

In two triangles, if two angles are equal to two angles, the remaining angles are also equal.

1-33, 1-34

1-34-conv

=1-4 + 1-27. If diameters of a quadrilateral bisects each other, the quadrilateral is parallelogram.

1-35, 1-37, 1-38, 1-39, 1-41

1-41-trap

2·trapezium = rec(sum of two bases, height)

1-44

1-44-x

Parabolic application of an area equal to a given rectilinear figure (not necessarily triangle)

1-46, 1-47

1-47-cor-1

In a rectangular triangle, the hypothenuse is greater than either of the remaining sides.

1-47-cor-2

In rectangular triangles of a given height, the hypothenuse is greater if the base is greater.

2-1, 2-2, 2-3, 2-4, 2-5

2-5-x

If C bisects AB and D cuts AB, r(AD, DB) < q(AC). (A special case of 6-27)

2-6, 2-8, 2-9, 2-10, 2-14

2-A

q(x+y)-q(x-y) = 2r(2x,y)

2-B

complicated "geometric algebra".

3-def-11

3-def-11-x

definition of similar arcs of a circle.

3-1, 3-3-Dir, 3-3-Conv

3-3-x

(see also 1-5-x) If from the center of a circle a perpendicular is drawn to a chord, and the extremities of the chord and the center is joined, there will be two congruent triangles.

3-7, 3-8

3-11

3-11-conv

If the straight line joining the centers of two circles pass the point of intersection, the circles will touch at this point (internally or externally)

3-11-x

(=3-11 + 3-16-Cor) If two circles touch one another internally, the tangents at their point of contact will coincide.

3-11-x-conv

(=3-11-conv + 3-16-Cor) If tangents of two circles at their point of intersection coincide, the circles will touch at this point. (internally or externally).

3-12-

included in 3-11.

3-15, 3-16-Cor, 3-17, 3-18, 3-19, 3-20, 3-21

3-21-conv

If C and D are on the same side of AB, and angle ACB is equal to angle ADB, then A, B, C, D are on a circle.

3-22

3-22-conv

If two opposite angels of a quadrilateral are equal to two right angles, the quadrilateral is inscribed in a circle.

3-22-x

An exterior angle of a quadrilateral inscribed in a circle is equal to the opposite interior angle.

3-22-x-conv

In a quadrirateral, if an exterior angle is equal to the opposite interior angle, then the quadrilateral is inscribed in a circle.

3-25, 3-26, 3-27, 3-28, 3-29, 3-31

3-31-conv

In a circle, if an angle at circumference is right, it is on a diameter.

3-32, 3-35

3-35-conv

If two finite straight lines cut each other, and the rectangles contained by the segments of the lines are equal, then the four extremities of the given lines lie on a circle.

3-36

3-36-cor-1

A tangent of a circle is longer than a line cutting the circle.

3-36-cor-2

If two tangents are drawn from one point to a given circle, the tangents are equal each other.

3-36-x

If two chords of a circle cut one another when prolonged, then the rectangles contained by the segments between the point of intersection and the circumference are equal one another.

3-36-x-conv

If two finite straight lines cut each other when prolonged, and the rectangles contained by the segments between the extremities of the line and the point of intersection are equal to one another, then the four extremities of the given lines lie on a circle.

3-37

4-5

5-4

5-7-A

If a=b then a:c::b:c.
Arguments of the type:
a:b::c:d and e=a —> e:b::c:d
is NOT counted, though strictly speaking 5-7-A and 5-11 is used.

5-7-B

If a=b then c:a::c:b�D(The same criterion as 5-7-A applies).

5-7-Cor

If magnitudes are proportional, they will also be proportional conversely.

5-7-Cor-ineq

(Mueller VM) If a:b>c:d then d:c>b:a

5-7-cor-1

(Mueller VD) If a=b and c=d, then a:b=c:d (See the introductory essay of this paper.)

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 b<c.

5-11

5-12, 5-12-ineq, 5-13, 5-13-ineq

5-14, 5-14-a (Mueller VE): See the introductory essay of this pape

5-15, 5-16, 5-17, 5-17-ineq

5-17-x

If a:b::c:d then b-a:b::d-c:d

5-18, 5-18-ineq, 5-19, 5-19-Cor, 5-19-Cor-ineq

5-19-x

If a:b=c:d then a-c:b-d=c:d (5-19 + 5-11)

5-22, 5-22-ineq, 5-23, 5-24, 5-25

5-B

(Mueller VB) It is impossible a:b=c:d and a:b>c:d at the same time.

5-P

See 308 (prop.231). Generalization of the manipulation of ratios such as componendo, dirimendo, enallax, etc.

5-Q

If a:b > c:d, then c:d < a:b.

6-def-1

Definition of similar rectilinear plane figures.

6-1

6-2-Dir

(Fig. 1) If k//l, then a:b::c:d.

6-2-Dir-eq

(Fig. 1) If k//l and a=b, then c=d.

6-2-Conv

(Fig. 1) If a:b::c:d, then k//l.

6-2-x

(Fig. 2, k//l) a:b::c:d.

6-2-x-eq

(Fig. 2, k//l) If a=b, then c=d.

6-2-x-conv

(Fig. 2, k//l) If a:b::c:d, then p, q, r meet at one point.

6-2-x2

(Fig. 3, k//l//m) a:b::c:d

6-2-x2-eq

(Fig. 3, k//l//m) If a=b, then c=d.

6-3-Dir, 6-3-Conv,

6-3-x, 6-3-x-conv

(for the line bisecting the external angle)

6-4

6-4-x

(Taisbak's theorem) = 6-4 + 6-17-Dir = 3-32-conv + 3-36

6-4-x-conv (converse of Taisbak's theorem) = 6-17-Conv + 6-6 = 3-37 + 3-32 see also 187-10 (p.253)

6-6, 6-7, 6-8, 6-8-conv, 6-8-Cor, 6-8-Cor-conv

6-8-x

If from the right angle A of a rectangular triangle ABC a perpendicular AD is drawn, then:
q(AD)=r(BD,DC),
q(AB)=r(CB,BD),
q(AC)=r(BC,CD).

6-10

6-12 (= Pappus 44)

6-12-x

To find a line d such that a:b::q(c):q(d) etc.

6-13

6-16-Dir, 6-16-Conv

6-16-ineq-dir

(Pappus 58) If a:b>c:d then r(a,d)>r(b,c)

6-16-ineq-conv

(Pappus 59) If r(a,d)>r(b,c) then a:b>c:d

6-17-Dir, 6-17-Conv

6-19, 6-20, 6-22, 6-22-Dir, 6-22-Conv

6-22-ineq

If a:b>c:d, then q(a):q(b)>q(c):q(d)

6-23

6-23-def

(a:b)(b:c)=a:c. Definition of compounded ratio.

6-23-lem

Uniqueness of compounded ratio:
if a:b=x:y and c:d=z:w, then (a:b)(c:d)=(x:y)(z:w)

6-23-lem-2

If (a:b)(c:d)=(x:y)(z:w), and a:b=x:y, then c:d=z:w.

6-27, 6-28, 6-29

6-29-cor

Uniqueness of the solution of excessive application of areas.

6-32

11-def-3, 11-def-4, 11-5, 11-11, 11-32, 11-37

• Propositions from Pappus' Book 7.

44

(= Euclid 6-12)

47, 52, 53, 62, 68, 70, 71, 73, 74, 79, 81, 83, 86, 89, 95, 96, 97, 113, 114, 114-conv, 127, 130, 142, 165, 167, 191, 196, 204, 205, 208, 210, 214, 241, 275, 286, 296, 297, 315, 320

• Definitions and propositions in Apollonius' Conics.

Con-1-def-2, Con-1-12, Con-1-21, Con-1-50, Con-1-52, Con-1-53, Con-1-54, Con-1-56, Con-2-1, Con-2-3, Con-2-8, Con-2-10, Con-2-14,

• Definitions and propositions of the Data.

Data-def-1, Data-def-2, Data-def-4, Data-def-6, Data-2, Data-3

Data-3-x

a, m given —> ma, a/m given

Data-4, Data-5

Data-6-x

(=5-17-Data) a+b:b or a+b:a given —> a:b given

Data-7, Data-8, Data-25, Data-26, Data-27, Data-28, Data-29, Data-30, Data-50, Data-52, Data-54, Data-55, Data-57, Data-59, Data-70, Data-91, Data-93

5-12-Data, 5-19-Data

Con-1-11-Data, Con-1-53-Data

• Other propositions (from Determinate Section, Menelaus' theorem)

DetSection, DetSection-1-3-2

Menelaus

(See Jones, p.457)

Pappus-6-12, Pappus-6-12-x

See Jones p.423 and Eecke 2:382 n.1.