The applications we have described are sufficient to show the variety of ways in which the circle is employed in the arts. The principal reason of its

frequent use is the facility with which it may be described, and the simplicit of the elements which determine it; namely the centre and the radius.

Questions for Examination.

1. What is meant by concentric circles?

2. Show how to describe a circumference of a given radius which shall be concentric with another circumference, the centre of which cannot be found.

3. Prove that the line joining the centres of two circles which intersect is perpendicular to their common chord, and bisects it. 4. Describe a circumference of given radius which shall cut another in two given points.

5. The point of contact of two circles which touch is on the line of centres.

6. Describe a circle of given radius

Ist. Touching exteriorly two given circles.

2nd. Touching interiorly two given circles.

3rd. Touching one circle interiorly and the other exteriorly. 7. Describe a circle of given radius

Ist. Touching another circle in a given point.

2nd. Touching a given circle, and passing through a given point.

3rd. Touching a circle and a given straight line.

8. Describe a circle

Ist. Touching another in a given point, and passing through a given point.

2nd. Touching another, and passing through two given points.

3rd. Passing through a given point, and touching a straight line and a given circle.

4th. Touching a given straight line and a given circle.

9. Describe the use and construction of toothed wheels. 10. What is meant by an oval, and what by a basket-handle arch?

II. Describe a basket-handle arch with five centres.

12. The same with nine centres.

13. The same with eleven centres.

14. Describe the oblique oval arch.

15. Describe the irregular oval.

16. Draw the Ionic volute, and explain fully the process. 17. Name other applications of circles in combination.

Theorems and Problems.

1. Two circles are given in position and magnitude. If two parallels be drawn, each touching one of the circles, the straight line which joins the points of contact passes through one of two fixed points.

2. A segment of a circle being given, describe the circle of which it is the segment.

3. If two straight lines cut one another within a circle, one of them passing through the centre, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other.

4. A ladder is gradually raised against a wall; find the locus of its middle point.

5. How would you practically determine, with a foot-rule only, whether an angle traced upon paper or on the ground, and said to be a right-angle, is really a right-angle?

6 Four points being given in a plane, how could you determine, before constructing a circle through any three of them, whether such a circle wouldpass through the fourth point?

7. To inscribe three equal circles in an equilateral triangle, each circle touching two of its sides.

8. To inscribe six equal circles in an equilateral triangle, each circle touching only one of its sides.

9. To inscribe four equal circles in a square, each circle touching two of its sides.

10. To inscribe three equal circles in a given circle.

II. The chord of a segment of a circle being given, and a tangent to the circle, describe the segment.

12. To inscribe four equal circles in a given circle.

13. To inscribe five equal circles.

14. To inscribe three semicircles having their diameters adjacent

Ist. In an equilateral triangle. 2nd. In a given circle.

15. To inscribe in a given square four equal semicircles having their diameters adjacent and their arcs touching two sides of the given square.

16. DF is a straight line touching a circle, and terminated by AD, BF, the tangents at the extremities of the diameter A B; show that the angle which D F subtends at the centre is a rightangle.

17. Describe a circle through a given point, and touching a given straight line, so that the chord joining the given point and point of contact may cut off a segment containing a given angle.

18. To describe a circle through two given points to cut a straight line given in position, so that a diameter of the circle drawn through the point of intersection shall make a given angle with the line.

19. Of two circular segments upon the same base, the larger is that which contains the smaller angle.

20. If any two circles, the centres of which are given, intersect each other, the greatest line which can be drawn through either point of intersection and terminated by the circles, is independent of the diameters of the circles.

21. ABD, AC B are the arcs of two equal circles cutting one another in the straight line A B ; draw any chord A CD cutting both arcs, and join C B, D B ; C B is equal to D B.

22. Draw through one of the points in which any two circles cut one another, a straight line which shall be terminated by their circumferences and bisected in their point of section.

23. Describe two circles with given radii which shall cut each other and have the line between the points of section equal to a given line.

24. Two circles cut each other, and from the points of intersection straight lines are drawn parallel to one another, the portions intercepted by the circumferences are equal.

25. A B D, AC B, are the arcs of two equal circles cutting one another in the straight line A B ; draw the chord A CD cutting the inner circumference in C and the outer in D, such that AD and D B together may be double of A C and C B together.

26. If from two fixed points in the circumference of a circle, straight lines be drawn intercepting a given arc and meeting without the circle, the locus of their intersections is a circle.

27. Show that, if two circles cut each other, and from any point in the straight line produced which joins their intersections two tangents be drawn, one to each circle, they shall be equal to one another.

28. If three equal circles have a common point of intersection, prove that a straight line joining any two of the points of intersection will be perpendicular to the straight line joining the other two points of intersection.

29. Two equal circles are drawn intersecting in the points A and B, a third circle is drawn with centre A and any radius not greater than AB intersecting the former circles in D and C. Show that the three points B, C, D, lie in one and the same straight line.

30. Through two given points to describe a circle bisecting the circumference of a given circle.

31. If two circles touch each other internally, prove that the straight lines which join the extremities (on the same side of the common diameter) of any two parallel diameters, pass through the point of contact.

32. A common tangent is drawn to two circles which touch each other externally; if a circle be described on that part of it which lies between the points of contact, as diameter, this circle will pass through the point of contact of the two circles, and will touch the line which joins their centres.

33. If two circles touch each other externally, and parallel diameters be drawn, the straight line joining the extremities of these diameters will pass through the point of contact.

34. If two circles touch each other, any straight line passing through the point of contact cuts off segments which contain equal angles.

35. If a circle roll within another of twice its size, any point in its circumference will trace out a diameter of the first.

36. Given two circles: it is required to find a point from which tangents may be drawn to each, equal to two given straight lines.

37. Shew that all equal straight lines in a circle will be touched by another circle.

38. Two circles are described about the same centre: draw a chord to the outer, which shall be divided into three equal parts by the inner one. How is the possibility of the problem limited? 39. The circles described on the sides of any triangle as diameters will intersect in the sides, or sides produced, of the triangle.

40. The circles which are described upon the sides of a rightangled triangle as diameters, meet the hypothenuse in the same point; and the line drawn from the point of intersection to the centre of either of the circles will be a tangent to the other circle.

Arithmetical Exercises.

1. Two circles intersect, and lines are drawn cutting them, one of which passes through the centres, and cuts the other in the middle point O of the chord of contact. The parts of the line not passing through the centres are E F=33 ft., FO=4}, OG =3, G H=3; and of the other, A B not measured, BO=3, OC=2, CD not measured. Find the lengths of the radii of the circles, and the chord of contact. Ans. 6 ft., 5 ft., 10 ft.

2. In a circle of radius 10 inches, a square is inscribed, and in the square a circle. Find the length of the side of the square and the radius of the inner circle.

Ans. 14 142136 in. and 7:071068 in. 3. A flag-staff, 45 ft. high, is erect on a tower 90 ft. high; at what point on the horizon will the flag-staff appear under the greatest angle? Ans. 110 227 ft. 4. A person observes the elevation of a tower to be 60°, and in receding from it 100 yards farther he finds the elevation to be 30°. Required the height of the tower. Ans. 86.602 ft. 5. An equilateral triangle is described having its angular points in the sides of a right-angled isosceles triangle, and one side parallel to the hypothenuse. The length of a side of the rightangle is 10 inches; find the the area of the equilateral triangle. Ans. 44 8038 sqr. in.

6. Two objects, A and B, were observed from a ship to be at the same instant in a line inclined at an angle of 15° to the east of its course, which was at the time due north. The ship's course was then altered, and after sailing five miles in a N.W. direction, the same objects were observed to bear E. and N.E. respectively. Required the distance of A from B.

Ans. 6 33974 miles.

CHAPTER XI.

SIMILAR FIGURES.

265. When a design is copied on a reduced scale by the methods described in § 185-9 the drawing produced is said to be similar to the