But Lafb= dfe, inasmuch as they are vertical s;

.. in As abf, dfe, there are 2 s of the one = 2 s of the other, each to each;

.. rem. abd rem. La ed.

M.-We have now arrived at the last case, "when the point of intersection of two chords falls within the circle." If the point of intersection is in the centre, what, then, are the chords?

P.-Diameters of the circle.

M.-And, if we conceive rectangles formed by their segments, what conclusion must necessarily follow respecting them?

P.-Such rectangles are squares, and are equal to each other.

M.-But, if the point of intersection be not the centre, are the segments, then, equal to each other? P.-No.

e

Join cf:
then, if bf=fa,
Lcfb is a rt. L;
and if effd,

a

.. ab and ed do not bisect each other.

M.-Again, if the point of intersection is not the centre of the circle, are, then, the rectangles contained by their segments equal to each other?

P.-Yes the rectangle contained by a e, e b, is equal to the rectangle contained by ce, ed.

Join a c and db:

:. La cd

similarly,

Lab d, as they are on the same arc a d; cab

cdb,

and cea bed, as they are vertical ≤s:

..A aec is similar to ▲ bed;

and.. ae ed ec:eb;

and, hence, a ex ebed x ec,

that is, rectangle a e, e b = rectangle e d, e c.

M.-There is a particular case depending on this truth.-Let one of the two straight lines be a diameter, and the other a chord at right angles to it: find how the truth, you have demonstrated, is then modified.

b

P.-The rectangle a e, eb

rectangle ce, e d.

But ce ed, because a b is at right angles to cd;

.. rectangle ce, e d = ce2 or de2;

and.. rectangle a e, e b c e2.

M.-Express this truth in words.

P.—If, from a point in the circumference of a circle, a perpendicular be drawn to the diameter, the rectangle contained by the segments of the diameter is equal to the square of the perpendicular.

SUBSTANCE OF SECTION I.

1. A circle has only one centre.

2. A diameter is the longest straight line that can be drawn in a circle.

3. An arc of a circle is a portion of its circumference.

4. The straight line joining the extremities of an arc is called a chord.

5. A perpendicular drawn from the centre to a chord bisects the chord.

6. If a chord be bisected, the straight line joining the point of bisection and the centre of the circle is perpendicular to the chord.

7. Arcs are the measures of the angles which they subtend at the centre.

8. Chords which are equi-distant from the centre of a circle are equal to each other.

9. Equal chords, in a circle, are equi-distant from the centre.

10. Chords, in a circle, which are not equi-distant from the centre are not equal to each other, and the lesser chord is farther from the centre than the greater chord.

11. The angle at the centre of a circle is double the angle at the circumference, upon the same arc. 12. At the circumference, an angle which stands upon a semi-circumference is a right angle.

13. At the circumference, an angle which stands upon an arc less than a semi-circumference is an acute angle; and that which stands upon an arc greater than a semi-circumference is an obtuse angle.

14. At the circumference, angles which stand upon the same arc are equal to each other.

15. In a circle, if two chords intersect each other, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.

16. If, from any point in the circumference, a perpendicular be drawn to a diameter, the rectangle contained by the segments of the diameter is equal to the square of the perpendicular.

SECTION II.

THREE AND MORE straight lines IN A CIRCLE

M-If, from any point in a circle, three straight lines be drawn to the circumference, what will result from comparing them?

P.-Nothing definite can be said of them, except when the point is the centre.

M.-But, if one of the three straight lines passes through the centre-?

P. The straight line which passes through the centre is the greatest of the three.

Let ab pass through the centre c; ab shall be greater than ad or a e.

Join cd, ce:

兒

d

then, c is the centre of the circle,

cb=cd=c e.

But ac+cd> ad;

.accbad;

that is, ab> ad.

Similarly, ab> ae;

..ab is the greatest straight line.