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### Table of contents

- Definitions (7)
- Propositions (16)

- Guide to Book IV
- Logical structure of Book IV

### Definitions

Definition 1.
  A rectilinear figure is said to be inscribed in a rectilinear figure when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed.

Definition 2.
  Similarly a figure is said to be circumscribed about a figure when the respective sides of the circumscribed figure pass through the respective angles of that about which it is circumscribed.

Definition 3.
  A rectilinear figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle.

Definition 4.
  A rectilinear figure is said to be circumscribed about a circle when each side of the circumscribed figure touches the circumference of the circle.

Definition 5.
  Similarly a circle is said to be inscribed in a figure when the circumference of the circle touches each side of the figure in which it is inscribed.

Definition 6.
  A circle is said to be circumscribed about a figure when the circumference of the circle passes through each angle of the figure about which it is circumscribed.

Definition 7.
  A straight line is said to be fitted into a circle when its ends are on the circumference of the circle.

### Propositions

Proposition 1.
  To fit into a given circle a straight line equal to a given straight line which is not greater than the diameter of the circle.

Proposition 2.
  To inscribe in a given circle a triangle equiangular with a given triangle.

Proposition 3.
  To circumscribe about a given circle a triangle equiangular with a given triangle.

Proposition 4.
  To inscribe a circle in a given triangle.

Proposition 5.
  To circumscribe a circle about a given triangle.

Corollary. When the center of the circle falls within the triangle, the triangle is acute-angled; when the center falls on a side, the triangle is right-angled; and when the center of the circle falls outside the triangle, the triangle is obtuse-angled.

Proposition 6.
  To inscribe a square in a given circle.

Proposition 7.
  To circumscribe a square about a given circle.

Proposition 8.
  To inscribe a circle in a given square.

Proposition 9.
  To circumscribe a circle about a given square.

Proposition 10.
  To construct an isosceles triangle having each of the angles at the base double the remaining one.

Proposition 11.
  To inscribe an equilateral and equiangular pentagon in a given circle.

Proposition 12.
  To circumscribe an equilateral and equiangular pentagon about a given circle.

Proposition 13.
  To inscribe a circle in a given equilateral and equiangular pentagon.

Proposition 14.
  To circumscribe a circle about a given equilateral and equiangular pentagon.

Proposition 15.
  To inscribe an equilateral and equiangular hexagon in a given circle.

  Corollary. The side of the hexagon equals the radius of the circle.

  And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle an equilateral and equiangular hexagon in conformity with what was explained in the case of the pentagon.

  And further by means similar to those explained in the case of the pentagon we can both inscribe a circle in a given hexagon and circumscribe one about it.

Proposition 16.
  To inscribe an equilateral and equiangular fifteen-angled figure in a given circle.

  Corollary. And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle a fifteen-angled figure which is equilateral and equiangular.

  And further, by proofs similar to those in the case of the pentagon, we can both inscribe a circle in the given fifteen-angled figure and circumscribe one about it.

### Guide to Book IV

All but two of the propositions in this book are constructions to inscribe or circumscribe figures.

Figure | Inscribe figure in circle | Circumscribe figure about circle | Inscribe circle in figure | Circumscribe circle about figure
Triangle | IV.2 | IV.3 | IV.4 | IV.5
Square | IV.6 | IV.7 | IV.8 | IV.9
Regular pentagon | IV.11 | IV.12 | IV.13 | IV.14
Regular hexagon | IV.15 | IV.15,Cor | IV.15,Cor | IV.15,Cor
Regular 15-gon | IV.16 | IV.16,Cor | IV.16,Cor | IV.16,Cor

There are only two other propositions. Proposition IV.1 is a basic construction to fit a line in a circle, and proposition IV.10 constructs a particular triangle needed in the construction of a regular pentagon.

### Logical structure of Book IV

The proofs of the propositions in Book IV rely heavily on the propositions in Books I and III. Only one proposition from Book II is used and that is the construction in II.11 used in proposition IV.10 to construct a particular triangle needed in the construction of a regular pentagon.

Most of the propositions of Book IV are logically independent of each other. There is a short chain of deductions, however, involving the construction of regular pentagons.

Dependencies within Book IV
1, 5 | 10
2, 10 | 11
11 | 12
1, 2, 11 | 16

Elements Introduction - Book III - Book V.

© 1996
D.E.Joyce
Clark University