The area of a sector BCAD is found by multiplying the length of the arc CAD (see the rule, page 171) by half the radius BD; or we may first find what part of the circumference the arc CDA is; whether a third, a fourth, a fifth, &c. and then di- D vide the area of the whole circle whose radius is BC, by 3, 4, 5, &c. according as the arc CDA is,, &c. of the whole circumference. If we are to find the area

A

B

of the segment CDA, we must first find the area of the sector BCDA; and then also the area of the triangle ABC; which, subtracted from the area of the sector BCDA, will leave the area of the segment CDA.

RECAPITULATION OF THE TRUTHS CONTAINED IN FOURTH SECTION.

Q. Can you now repeat the different relations, which exist between the different parts of a circle and the straight lines, which cut or touch the circumference?

A. 1. A straight line can touch the circumference in only one point.

2. When the distance between the centres of two circles is less than the sum of their radii, the two circles cut each other.

3. When the distance between the centres of two circles is equal to the sum of their radii, the two circles touch each other exteriorly.

4. When the distance between the centres of two circles is equal to the difference between their radii, the two circles touch each other interiorly.

5. When two circles are concentric, that is, when they are both described from the same point as a centre, the circumferences of the two circles are parallel to each other.

6. A perpendicular, dropped from the centre of a circle upon one of the chords in that circle, divides that chord into two equal parts.

7. A straight line, drawn from the centre of a circle to the middle of a chord, is perpendicular to that chord.

8. A perpendicular drawn through the middle of a chord, passes, when sufficiently far extended, through the centre of the circle.

9. Two perpendiculars, each drawn through the middle of a chord in the same circle, intersect each other at the centre.

10. The two angles, which two radii drawn to the extremities of a chord, make with the perpendicular dropped from the centre of the circle to that chord, are equal to one another.

11. If two chords, in the same circle, or in equal circles, are equal to one another, the arcs subtended by them are also equal; and the reverse is also true; that is, if the arcs are equal to one another, the chords which subtend them are also equal.

12. The greater arc stands on the greater chord, and the greater chord subtends the greater arc.

13. The angles at the centre of a circle are to each other in the same ratio, as the arcs of the circumference intercepted by their legs.

14. If two angles at the centre of a circle are equal to one another, the arcs of the circumference intercepted by their legs are also equal; and the reverse is also true, that is, if the two arcs intercepted by the legs of two angles at the centre of a circle, are equal to one another, these angles are also equal.

15. Angles are measured by arcs of circles, described with any radius between their legs. The circumference is for this purpose divided into 360 equal parts, called degrees; each degree into 60 equal parts, called minutes; each minute again into 60 equal parts, called seconds, &c.

16. The magnitude of an angle does not depend on the greatness of the arc intercepted by its legs; but merely on the number of degrees, minutes, seconds, &c. it measures of the 'circumference.

17. The circumference of a circle is the measure of 4 right angles; the semi-circumference that of 2 right angles; and a quadrant that of 1 right angle.

18. A straight line, drawn at the extremity of the diameter or radius perpendicular to it, touches the circumference only in one point, and is therefore a tangent to the circle.

19. A radius or diameter, drawn to the point of tangent, is perpendicular to the tangent.

20. A line, drawn through the point of a tangent perpendicular to the tangent, passes, when sufficiently far extended, through the centre of the circle.

21. The angle, formed by a tangent and a chord, is half of the angle at the centre, which is measured by the arc subtended by that chord; therefore the angle, formed by the tangent and the chord, measures half as many degreés, minutes, seconds, &c. as the angle at the centre.

22. The angle which two chords make at the circumference of a circle, is half of the angle made by two radii at the centre, if the legs of both these angles stand on the extremities of the same arc; therefore every angle, made by two chords at the circumference of a circle, measures half as many degrees, minutes, seconds, &c. as the arc intercepted by its legs.

23. If several angles at the circumference have their legs standing on the extremities of the same arc, these angles are all equal to one another.

24. Parallel chords intercept equal arcs of the circumference.

25. If from a point without the circle you draw a tangent to the circle, and, at the same time, a straight line cutting the circle, the tangent

is a mean proportional between that whole line, and the part of it which is without the circle.

26. If a chord cuts another within the circle, the two parts, into which the one is divided, are in the inverse ratio of the two parts, into which the other is divided.

27. If from a point without a circle, two straight sides are drawn, cutting the circle, these lines are to each other in the inverse ratio of their parts without the circle.

28. If the circumference of a circle is divided into 3, 4, 5, &c. equal parts, and then the points of division are joined by straight lines, the rectilinear figure, thus inscribed in the circle, is a regular polygon of the same number of sides, as there are parts into which the circumference is divided.

29. If from the centre of a regular polygon inscribed in a circle, radii are drawn to all the vertices at the circumference, the angles which these radii make with each other at the centre, are all equal to one another.

30. The side of a regular hexagon inscribed in a circle is equal to the radius of the circle.

31. If from the centre of a circle radii are drawn, bisecting the sides of a regular inscribed polygon, and then at the extremities of these radii tangents are drawn to the circle, these tangents form with each other a regular circumscribed polygon of the same number of sides as the regular inscribed polygon.