draw ac to the centre of the circle a df, draw a b at right angles to a c, .ba touches the circles af d and a eg, and .. a c, produced, passes, likewise, through the centre, h, of the circle a e g. Moreover, these circles cannot have another point of contact besides the point a; for, a straight line drawn from any other point k, in a b, through c, cannot likewise pass through the point m. Hence it follows, that, if two circles touch each other internally, the straight line joining their centres passes through the point of contact. Obs.--Most of the preceding truths may be considered almost as axioms; and, like all axioms, they may be clearly established, by assuming the contrary to be true, and, then, showing the absurdity resulting from the supposition. In order to show, for instance, that two circles, which cut each other, cannot cut each other in more than two points, the contrary may be assumed,—namely, f d that the circumference abce can cut the circum ference abcf in more than two points,-in a, b, c. Let d be the centre of the circle a bef, and.. the point d is, likewise, the centre of the circle abce: but, two circles which cut each other cannot have the same centre; therefore, the assumption (that two circles can cut each other in more than two points,) is erroneous; and, since it is an error to assume any number of points except two, it follows, that two circles can cut each other in two points only. It may be, here, remarked that, as an useful exercise, the master should require the pupils, to draw lines, in certain directions, in circles which intersect or touch each other, and, then, to discover such truths as necessarily follow therefrom. An exercise of greater importance, however, is the application of the properties of the circle to the construction of rectilineal figures. The best course of proceeding, in this and every case, is that in which the subject naturally presents itself, of which the following outlines are offered for the guidance of the master. It is required, I. STRAIGHT LINES. 1. To draw a straight line equal to a given straight line. 2. To draw two, three, four, &c. straight lines equal to a given straight line. 3. To bisect a given finite straight line. 4. To draw a straight line at right angles to a given straight line. 5. To draw a straight line at right angles to a given straight line from a given point in the same. 6. To draw a straight line at right angles to a straight line from a given point without it. 7. To draw a straight line parallel to a given straight line. 8. From a given point, to draw a straight line parallel to a given straight line. It is required, II.-ANGLES. 1. To make a right angle. 3. To make an acute angle. 4. To make an angle equal to a given angle. 5. To bisect a given angle. It is required, III. TRIANGLES. 1. To describe a right-angled triangle. 4. To make a given finite straight line the side opposite to the right angle,* in a right-angled triangle. 5. To make a given finite straight line one of the sides containing the right angle, in a right-angled triangle. * Hypotenuse [otívovca, Gr.]. 6. To describe an equilateral triangle. 7. Upon a given finite straight line, to describe an equilateral triangle. 8. To describe an isosceles triangle. 9. To make a given finite straight line the base of an isosceles triangle. 10. What requisites must three straight lines have, in order to become the sides of a triangle? 11. With three given straight lines, (under the restriction in question,) to construct a triangle. IV. QUADRILATERAL FIGURES. It is required, 1. To describe a square. 2. To describe a rhomb. 3. To describe a rectangle. 4. To describe a parallelogram. 5. Upon a given finite straight line, to describe a square. 6. To make a given finite straight line the diagonal of a square. 7. To make a given straight line the diagonal of a rhomb,-of a rectangle,-of a parallelogram. V.-POLYGONS. It is required, 1. To describe a pentagon [not an equilateral and equiangular pentagon].* * This problem is proposed, now, in order to stimulate inquiry : it solution should be learnt from Euclid. 2. To describe a regular hexagon. 3. Upon a given straight line, to describe a regular hexagon. VI.-CIRCLES. It is required, 1. To find the centre of a given circle. 2. [What requisite must a given straight line have in order to be a chord in a given circle?] 3. With the restriction in question, to place a given straight line in a given circle. 4. To place a right angle in a given circle so that its vertex may be in the circumference of the circle.' 5. To describe a circle about a right-angled triangle. 6. To describe a circle about an obtuse-angled triangle. 7. To describe a circle about an acute-angled triangle. 8. At a given point in the circumference of a circle, to draw a tangent. 9. From a given point without a circle, to draw a tangent. 10. In a right-angled, obtuse-angled, and acuteangled triangle, to inscribe a circle. 11. In a given circle, to inscribe a square. 12. About a given circle, to describe a square. |