Nor can any third line (as PZ) be drawn so that

For however it is drawn it will subtend a different angle at O from PX and PY.

And.. (y) will be greater or less than PX and PY.

K

Note-It is customary in part (y) of Prop. 7, as well as in the corresponding part of Prop. 8, to say-that which is nearer the line through the centre is greater than the one more remote-It is difficult to see how, of three concurrent lines, one can be properly said to be nearer to another than the third, without some arbitrary definition of the sense in which the word 'nearer' is used—as, for example, that it is to mean-more nearly coincident with. To avoid this difficulty the wording of the enunciation has been changed.

Proposition 8.

THEOREM-If from a point outside a circle straight lines are drawn to meet its circumference—

(a) the greatest is that which goes through the centre ; (B) the least is that which would, if it was produced, go through the centre;

(y) of any two which are incident on the concavity of the circumference, or of any two which are incident on the convexity, that one is the greater which subtends the greater angle at the centre;

(8) any one of the lines, excepting the greatest and least, will have one other line equal to it, but not more than one.

P

Let P be a pt. outside a O,

centre is O.

whose

Of lines drawn to the circumference from P

(a) let POA be the one through O, and PX any other. Join OX.

Then, in either case, since in As POX, POY, we have

Nor can any third line (as PZ) be drawn so that

PZ = PX = PY.

For however it is drawn, it will subtend a different Ʌ at O from PX or PY.

And.. by (y) will be greater or less than PX and PY.

Proposition 9.

THEOREM-If more than two equal lines can be drawn from a point within a circle to the circumference, that point is the centre.

For if one line is drawn from a point, within a circle but not the centre, to the circumf., only one other can be drawn from the point so that the two may be equal.

.. if three equal lines can be drawn from a point to circumf. that point must be the centre.

Proposition 10.

THEOREM-Two circles (which do not coincide) cannot have more than two points in common.

A

B

Let O be the centre of a O.

Then if A, B, C are three pts. on its circumf., they are equidistant from O.

.. if A, B, C could be on circumf. of another O, O would be the centre of this other ;

and then two concentric Os would cut :

which cannot be.

unless the Os coincide, they cannot have three pts. in

common.

Note-The arrangement of the enunciation and proof, given above, avoids the awkwardness of endeavouring to draw the impossible figure of two concentric circles, cutting in more than two points.