If two circles touch one another internally, they have not the same centre.

This proposition may be proved exactly as the former, (from def. 15, b. 1.)

PROP. 7. THEOR.

If from any point within a circle, which is not the centre, right lines be drawn to the circumference, the greatest is that which passes through the centre. The remaining part of the diameter is the least. Those lines which make equal angles with the diameter are equal.

That line which is nearer to that passing through the centre, is greater than one more remote.

And more than two right lines cannot be drawn, which shall be equal.

PART 1. The line passing through the centre is greater than any other drawn from this point.

Draw a right line from the centre to the extremity of the line not passing through the centre. This drawn line with the intercept between the centre and given point, are together greater than the line not passing through the centre, (prop. 20, b. 1,) but this drawn line is to the remaining part of the line passing through the centre, (def. 15, 1,) the line passing through the centre is greater than the other, &c. &c.

PART 2. The other part of the diameter is less than any other line drawn from this point.

Draw a right line from the centre to the extremity of the line that is not a part of the diameter. Then this drawn line is less than the sum of the intercept between the centre and given point and the other line, but the intercept and remaining part of the diameter are together

to the connecting line, (def. 15, b. 1,) .. this remaining part of the diameter, is less than any other line drawn from the given point.

PART 3. The right lines which at the same point make angles with the diameter are,

For if possible let one of them be greater than the other, cut from it (adjacent to the given point) a part to the other, join the centre and the extremities of those lines, produce the line joining the centre and cut of part, to the circumference.

Then there are two triangles formed, having the cut off part and intercept in one respectively to the other given line and intercept in the other, and the contained angles

(by hypoth.) .. the third sides are, viz. the joining lines, but one of these is also to the other produced to the circumference, (de 15, 1,).. a part of the produced line is to the whole, which is absurd, .. those lincs making = angles are not unequal.

PART 4. The line which is nearer to that passing through the centre, is greater than the one more reinote,

If the given lines be at the same side of the line passing through the centre, join the centre and their extremities, then there are two triangles formed having two sides of the one, (viz. the intercept and connecting line,) respectively to two sides of the other, (viz. the intercept and the other connecting line,) but in those triangles the angle subtended by the line nearest to that passing through the centre, is greater than that subtended by the other,.. (prop. 24, b. 1,) the side subtending the greater angle, viz. that nearest to the line passing through the centre, is greater than that subtending the lesser angle, viz. the more remote.

But if the given lines be at different sides, draw from the given point a right line, making with the least part of the diameter, an angle to that which the line at the other side makes with it, this line shall be to the given line on the other side, and is evidently greater than the other by the foregoing demonstration.

=

PART 5. More than two right lines cannot be drawn which shall be equal.

For let any three right lines be drawn from the given point to the circumference, and either one of them shall be part of the diameter, and greater or less than either of the others, (by part 1 and 2,) or two of them must be at the same side of the diameter, and.. unequal (by part 4.)

PROP. 8. THEOR.

If from any point without a circle, lines be drawn to the circumference, those which make equal angles with the line passing through the centre are equal.

Of those lines which are incident upon the concave circumference, the greatest is that which passes through the centre.

Of the rest, that which is nearest to the line passing through the centre, is greater than the more remote. But of those incident upon the convex ciscumference, that line is the least, which if produced, would pass through the centre.

Of the rest, that which is nearest to the least, is lesa than the more remoțe.

Only two lines can be drawn either to the concave or convex circumference which shall be equal.

PART 1. The right lines, which drawn from the given point make angles with that passing through the centre, are.

For if possible let one of them be greater than the other, cut from it a part (adjacent to the given point) = to the less, join the centre and the extremities of those = parts.

=

Then there are two triangles formed, having two sides of the one respectively to two sides of the other, and the contained angles (by hypoth.) .. the third sides or joining lines are, but of these the one joining the centre and uncut side is to the other when produced to the circumference, (def. 15, b. 1,) .. that joining the centre and cut line is to itself when produced to the circumference, a part to the whole, which is absurd. Therefore neither of the given lines is greater than the other, they

are .

PART 2. Of those lines which are incident upon the concave circumference, that line which passes through the centre is greater than any other,

Join the centre and the extremity of that not passing through the centre. Then this joining line is to that part of the-line passing through the centre, between the centre and circumference; add to both the remainder of the line passing through the centre, then the added part and joining line are together to the whole line passing

through the centre, but the former are greater than the line not passing through the centre,.. the line passing through the centre is greater than the other.

PART 3. The line which is nearer to the greater, is greater than the more remote.

If the given lines be at the same side of that passing through the centre, join the centre and their extremities at the circumference, then there are two triangles formed, having a joining line and common side in one respectively

to the other joining line and common side in the other, but of the angles contained by those sides, that subtended by the line nearest to that passing through the centre, is greater than that subtended by the more remote,

(by prop. 24, b. 1,) the line nearest to that passing through the centre, is greater than the more remote.

But if the given lines be at different sides, make on either sides, at the given point and with the line passing through the centre, an angle to the angle made by the given line at the other side; the line making this angle when produced to the concave circumference, shall be to the line at the other side. Then the proposition may be proved as in the foregoing case.

PART 4. Of those lines incident on the convex circumference, that line which if produced would pass through the centre, is less than any other.

Produce that one that would pass through the centre, to the centre, and connect the extremity of the other which is at the circumference with the centre.

Then this connecting line is to the produced part, but the connecting line with the connected line are together greater than the whole produced line, (prop. 20, b. 1,).. if the parts be taken away, the line which if produced would pass through the centre, is less than any other.

PART 5. The line which is nearer to the least, is less than the more remote.

If the given lines be at the same side of the least, join the centre and their extremities at the circumference. Then it is evident (from prop. 21, b. 1,) that the line next the least is less than the more remote, for the joining lines are, (def. 15, b. 1.)

But if the given lines be at different sides of the least, draw at one side to meet the convex circumference, a right line, making at the given point and with the least, an angle to that made by the line at the other side, then proceed as in the foregoing case.

PART 6. Only two equal right lines can be drawn either to the concave or convex circumference.

If any three lines be drawn, either one of them shall pass through the centre, and.. be either greater or less than the others, (by part 3,) or two must be at the same side of the line passing through the centre, and .'. unequal.

Schol. It is evident that any right line drawn to the convex circumference, is less than any right line drawn to the concave; it follows that if any three lines be drawn from a point without a circle to its circumference, only two of them can be equal.

Cor. Hence, and from part 5, prop. 7, it is evident that there is no point except the centre, from which three equal right lines can be drawn to the circumference of a circle.

PROP. 9. THEOR.

If a point be taken within a circle, from which more than two equal right lines can be drawn to the cir cumference, that point is the centre of the circle.

For if it were a point different from the centre, only two equal right lines could be drawn from it to the cir cumference.

PROP. 10, THEOR.

One circle cannot cut another in more than two points.

For if it be possible let it cut the other in three points, connect the centre of one of the circles with those points, the connecting lines are, (def. 15, b. 1,) but as the circles intersect they have not the same centre, (prop, 5, b. 3,) ... this centre is not the centre of the other circle, and .. as three right lines are drawn from a point not the centre, they are not, but it was shewn before that they were, which is absurd; the circles ... do not intersect in three points.

Schol. Hence it is evident that one circle cannot meet an other in more than two points.