or six chords, and consequently six different figures. But there is a distinction to be made between them, for in joining every third point we get the ordinary pentagon; every fifth, the equilateral triangle; and every sixth, the star pentagon. The alternate, the fourth, and the seventh points, give us quindecagons. These are the three numbers which are prime to the number of the sides of the given polygon. There are three star quindecagons, making with the original polygon in all four quindecagons.

There are therefore as many regular quindecagons plus one as there are numbers prime to 15 and less than its half. Five equilateral triangles, having their vertices at the points of division of a circumference divided into fifteen equal parts, form a figure having fifteen angles and fifteen sides, but not a star polygon.

The rule just enunciated is applicable in all cases. It shows us that since 2, 3, 4, 5, 6, are prime to 13 and less than its half, there are six regular polygons of thirteen sides, five of which are star polygons; there are three of sixteen sides, four of thirty sides, and so

on.

In order to construct the star polygon it is therefore sufficient to join, in any order, the angles of the given polygon, provided that the chord always cuts off the same number of angles, this number being determined beforehand by an examination similar to that just made.

To draw by extension a regular star polygon. 310. Produce the sides of the polygon so that the

Q

lines which intersect include one, two, three, etc., sides, and regular star polygons will be formed. We shall see that these polygons are similar to those formed by reduction, and are the same in number.

This results from the consideration of the following proposition:

311. The vertices of the outer angles of a regular star polygon constructed by extension, are those of a regular polygon of the same name.

It will easily be perceived (fig. 330) that these vertices are equidistant from the centre of the given polygon, and also that the angle formed by the two radii which join two successive vertices to the centre is constant and equal to the angle at the centre of the given polygon. Therefore the angles of the large star polygon are those of a polygon of the same name. Wherefore: The star polygon drawn by extension may be considered as deduced by reduction from another polygon of the same name.

In the same manner it may be shown that :The vertices of the inner angles of a regular star polygon constructed by reduction, are also those of a regular polygon of the same name.

Whence it follows that the star polygon drawn by reduction may be considered as deduced by extension from this other polygon.

The general conclusion to be drawn is that :—

312. A regular star polygon may be considered as deduced by extension or reduction from a regular polygon of the same name.

Thus the star pentagon ABCDE (fig. 324), may be considered as deduced by extension from the ordinary pentagon adbec, or by reduction from the pentagon ADBEC.

Every regular star polygon has the same symmetrical axes as the regular ordinary polygon from which it has been deduced.

313. From each apex of a regular ordinary polygon are drawn two sides of the star polygon, which are deduced either by reduction or extension (figs. 324 and 330), and these two sides form equal angles with the diameter drawn to this apex, which is a symmetrical axis of the ordinary polygon.

The star pentagon is the simplest of regular star polygons.

314. For the triangle and square do not form star

Questions for Examination.

1. Define a regular polygon and a regular star polygon. 2. Prove that the sum of the angles of a polygon is as many times two right-angles as the figure has sides less two.

3. Every regular polygon can be inscribed in or described about a given circle.

4. Inscribe in a given circle,-
Ist. An equilateral triangle.
2nd. A regular hexagon.
3rd. A regular dodecagon.
4th. A regular square.
5th. A regular octagon.

6th. A regular pentagon. 7th. A regular decagon. 8th. A regular quindecagon. 9th. A regular heptagon. 10th. Any regular figure.

5. A diameter through an angular point is an axis of symmetry.

6. A diameter through the middle point of a side is an axis of symmetry.

7. A polygon with an even number of sides has a centre of symmetry, but one with an odd number of sides has not.

8. Find the number of degrees in the angle at the centre and interior angle of any polygon.

9. Prove that if the radius be divided in extreme and mean ratio the greater segment is a chord of an arc of 36°. 10. Describe the graphometer.

II. Show how surfaces may be decorated,-
Ist. With equilateral triangles alone.
2nd. Squares alone.

3rd. Hexagons.

4th. Hexagons and equilateral triangles.
5th. Octagons and squares.

6th. Hexagons and rhombuses.

7th. Pentagons and rhombuses.

12. Show how to construct a rose wheel by rhombuses.
13. Explain what is meant by the involute to the circle.
14. Define a spiral, and give six ways of drawing it.
15. To draw a regular star polygon,-

Ist. By extension.

2nd. By reduction.

Theorems and Problems.

1. Trisect a given circle by dividing it into three equal

sectors.

2. The centre of the circle inscribed in and circumscribed about an equilateral triangle coincide; and the diameter of one is twice the diameter of the other.

3. If an equilateral triangle be inscribed in a circle, and a straight line be drawn from the vertical angle to meet the circumference, it will be equal to the sum or difference of the straight lines drawn from the extremities of the base to the point where the line meets the circumference, according as the line does or does not cut the base.

4. Prove that the perpendicular from the vertex on the base of an equilateral triangle is equal to the side of an equilateral triangle inscribed in a circle whose diameter is the base.

5. If an equilateral triangle be inscribed in a circle, and the adjacent arcs cut off by two of its sides be bisected, the line joining the points of bisection shall be trisected by the sides.

6. The perimeter of an equilateral triangle inscribed in a circle is greater than the perimeter of any other isosceles triangle inscribed in the same circle.

7. Prove that the area of a regular hexagon is greater than that of an equilateral triangle of the same perimeter.

8. Determine the distance between the opposite sides of a regular hexagon inscribed in a circle.

9. Inscribe a regular hexagon in a given equilateral triangle. 10. To inscribe a regular dodecagon in a given circle, and show that its area is equal to the square of the side of an equilateral triangle inscribed in the circle.

11. The locus of the centres of the circles which are inscribed