An ellipse is a plane figure bounded by one continuous curve described about two points, so that the sum

Fig. 11.

Fig. 12.

E

of the distances from every point in the curve to the two foci may be always the same-Fig. 11.

PROPERTIES OF THE CIRCLE.

A circle contains a greater area than any other plane figure bounded by the same length of circumference or outline.

A circle is a plane figure contained by one line and is such that all straight lines drawn from a point within the figure to the circumference are equal, and this point is called the center of the circle.

A diameter of a circle is a straight line drawn. through the center and terminated both ways by the circumference, as AC in Fig 12.

A radius is a straight line drawn from the center to the circumference as LH in Fig. 12.

A semicircle is the figure contained by a diameter and that part of circumference cut off by a diameter as AHC in Fig. 12.

A segment of a circle is the figure contained by a straight line and the circumference which it cuts off as DHE in Fig. 12.

A sector of a circle is the figure contained by two straight lines down from the center and the circumference between them as LMC in Fig. 12.

A chord is a straight line, shorter than the diameter, lying within the circle, and terminated at both ends by the circumference as DE in Fig. 12.

An arc of a circle is any part of the circumference as DHE in Fig. 12.

The versed sine is a perpendicular joining the middle of the chord and circumference, as GH in Fig. 12. Multiply the diameter by 3.1416

Circumference.

the product is the circumference.

Diameter. Multiply the circumference by .31831, the product is the diameter, or multiply the square root of the area by 1.12837, the product is the diameter. Area. Multiply the square of the diameter by .7854, the product is the area.

Fig. 13.

Side of the square. Multiply the diameter by .8862, the product is the side of a square of equal area. Diameter of circle. Multiply the side of a square by

1.128, the product is the diameter of a circle of equal

area.

To find the versed sine, chord of an arc or the radius when any two of the three factors are given.—Fig. 13.

To find the length of any line perpendicular to the chord of an arc, when the distance of the line from the

Fig. 14.

center of the chord, the radius of the arc and the length of the versed sine are given-Fig. 14.

To find the diameter of a circle when the chord and versed sine of the arc are given.

To find the length of any arc of a circle, when the

chord of the whole arc and the chord of half the arc are given-Fig. 15.

DEFINITION OF POLYGONS.

A polygon, if its sides are equal, is called a regular polygon, if unequal, an irregular polygon.

A pentagon is a five-sided figure.

A hexagon is a six-sided figure-Fig. 16.
A heptagon is a seven-sided figure.

An octagon is an eight-sided figure—Fig. 17.
A nonagon is a nine-sided figure.

A decagon is a ten-sided figure.

A unadecagon is an eleven-sided figure.
A duodecagon is a twelve-sided figure.

GEOMETRICAL DEFINITION OF SOLID FIGURES.

A solid has length, breadth and thickness. The boundaries of a solid are surfaces.

A solid angle is that which is made by two or more

plane angles, which are not in the same plane, meeting at one point.

A cube is a solid figure contained by six equal squares-Fig. 18.

Fig. 18.

Fig. 19.

A prism is a solid figure contained by plane figures of which two that are opposite are equal, similar, and parallel to one another, the other sides are parallelograms-Fig. 19.

A pyramid is a solid figure contained by planes, one of which is the base, and the remainder are triangles, whose vertices meet a point about the base, called the vertex or apex of the pyramid-Fig. 20.

Fig. 20.

Fig. 21.

A cylinder is a solid figure described by the revolution of a rectangular or parallelogram about one of its sides-Fig. 21.