PROP. 11. THEOR.

If two circles touch one another internally, the right line joining their centres being produced shall pass through a point of contact.

For if possible let the centres be so situated, that the right line connecting them will not pass through a point of contact; produce it both ways to meet the circumferences, and connect the point of contact with the

centres.

Then in the triangle formed by those connecting lines, the lines connecting the centres and that of the internal circle with a point of contact, are together greater than the line connecting the centre of the external circle with this point of contact, (prop. 20, b. 1,).. than the radius of the external circle which passes through the centre of the internal; let the line connecting the centres be taken from both, and the radius of the internal circle shall be greater than the remaining part of that of the external, of which it is a part, which is absurd,.. &c. &c.

PROP. 12. THEOR.

If two circles touch one another externally, the right line joining their centres shall pass through a point of contact.

For if possible let the centres be so situated that the right lines joining their centres does not pass through & point of contact; join the centres and a point of contact.

Then in the triangle formed by the three connecting lines, those drawn from the centre to this point of contact, are together greater than that joining the centres, but they are also to part of it, which is absurd, ... &c. &c.

PROP. 13, THEOR.

One circle cannot touch another externally or internally in more points than one.

If possible let two circles touch one another internally in two points; join the centres and produce the

joining line to one point of contact; draw right lines from the centres to the other point of contact. Then the right lines drawn from the centre of the lesser circle to the points of contact are, (def. 15, b. 1,) add to both the right line joining the centres, then the right line joining the centres and a radius of the lesser circle are together to a radius of the greater, but they are also greater than it, (prop. 20, b. 1,) which is absurd.

Next, if possible, let two circles touch one another externally in two points; draw a right line joining the centres and passing through one point of contact, and draw right lines from the centres to the other point of contact.

This then is evidently an absurdity, from def. 15, b. 1, and prop. 20, b. 1.

Therefore there is no case in which two circles can touch one another in two points.

Schol. Hence it is evident that a circle touching a circle never again meets it.

PROP. 14, THEOR.

In a circle equal right lines are equally distant from

the centre.

And right lines which are equally distant from the centre are equal.

Join the centre and either extremity of each given line, and draw lines from the centre at right angles to the given lines.

PART 1. Then because the given lines are and the perpendiculars from the centre bisect them, (prop. 3, b. 3,) half of one of them is to that of the other, .. thers of those halves are, but the right lines from the centre to the extremities being also their rs are , but the ☐ of each of those is to a 2 of an half and 2 of a perpendicular, (prop. 47, b. 1,).. the 'rs of the perpendiculars are, and .. the perpendiculars themselves, (cor. 2, prop. 47, b. 1.)

PART 2. Because the perpendiculars are their rs are, and because the lines from the centre to the extremities are their 2rs are, but each of the latter O'rs is to a 3 of a perpendicular with a 2 of half the line to which they are drawn,.. the ' of half one

2

given line is themselves are

to a of half the other, .. the halves and .'. the given lines are =.

PROP. 15. THEOR.

The diameter is the greatest right line in a circle, and of all others that which is nearer to the centre is greater than the more remote.

PART 1. The diameter is greater than any other line. For connect the centre with both extremities of the line not passing through the centre; those connecting lines are together to the diameter, (def. 15, b. 1,) but they are greater than the line not passing through the centre, (prop. 20, b. 1,) ... &c. &c.

PART 2 That which is nearer to the centre is greater than the more remote.

First let the given lines be on the same side of that passing through the centre and not intersecting one and other; connect the centre with the extremities of each of them.

Then in the triangles formed by the given lines and those connecting lines, the lines connecting the centre and extremities of the one nearest to the centre are = to those connecting the centre and extremities of the other, but the angle contained by the former is greater than that contained by the latter, .. that line subtending the greater angle, viz. that nearest to the one passing through the centre, is greater than the line subtending the smaller angle, viz. the more remote.

Let the given lines be at different sides or let them intersect. Draw perpendiculars to them from the centre, and from the greater perpendicular cut off a part to the less, and through the point of section draw a line at right angles to this perpendicular: this drawn line is to that on which the lesser perpendicular falls, (prop. 14, b. 3,) but it is greater than that on which the lesser perpendicular falls, (by the foregoing case,) ... &c. &c.

PROP. 16. THEOR.

The right line drawn from the extremitity of the dia meter of a circle perpendicular to it falls without the

circle.

And if any right line be drawn from any point within that perpendicular to the point of contact, it cuts the circle.

PART 1. For if it be possible, let there be a right line which meets it again and is perpendicular to the diaimeter, connect the centre and this point of contact.

Then because in the triangle thus formed, two of the sides are, being radii of the same circle, .. the angles at the points of contact are, but one of them is a right angle, which is absurd, ... &c. &c.

PART 2. If possible let a right line be drawn from a point between a perpendicular and circumference to the extremity of the diameter which does not cut the circle.

The angle contained by this drawn line and the diameter, must be less than a right angle; draw from the centre a line at right angles to it; this line drawn at right angles is less than a radius of the circle, (prop. 19, b. 1,) but it is also greater than it, which is absurd,.. &c. &c.

Schol. 1. Hence it is evident that the line at right angles to the extremity of the diameter, touches the circle in that point, and that the tangent can only meet the circle in one point, and that at each point of the cir cumference there is only one tangent.

Schol. 2. It is also evident that the right line, which makes at the extremity of the diameter an acute angle, however great, must meet the circle again.

Cor. 1. From this proposition is immediately deduced a method of drawing a tangent through any given point in the circumference of a circle: draw through the given point a diameter, and erect at the extremity of it a perpendicular.

Cor. 2. If the radius drawn to the given point in the circumference be produced beyond its extremity at the centre, and if there be taken in the produced part any number of points, and from these points as centres, circles be described through the given point, it is evident that each of those circles touch the tangent in that point, whence that part of the right line (drawn from the centre to an other point in the tangent) which lies between the circumference and tangent is infinitely divisable..

PROP. 17. PROB,

From a given point without a given circle, to draw a right line which shall be a tangent to the circle.

From the centre of the given circle as a centre and the given point as distance, describe a circle, connect the centre and given point; from the point where this connecting line cuts the given circle draw a line at right angles to it, to meet the circumference of the described circle, connect the point in which it meets it with the centre, and connect the point in which this connecting line cuts the given circumference with the given point, this line shall be the required tangent.

For thus there are two triangles formed having two sides of one respectively to two sides of the other, (viz. radii of the circles,) and the contained angle common, .*. the other sides and angles shall be respectively, but of these the angle contained by a radius of given circle, (viz. part of the line connecting the centre and given point,) and base is a right angle, (by constr.) .. the angle contained by the other radius and base is a right angle, but this base is the right line drawn from the given point to the circumference of the given circle and is .. a tangent to it.

Schol. It is evident that there can be drawn two tangents from the given point, one at each side of the line connecting the centre and given point.

PROP. 18. THEOR.

If a right line be a tangent to a circle, the right line drawn from the centre to the point of contact is perpendicular to it,

For if possible let any other right line from the centre be at right angles to it; then in the triangle formed by those three lines, (viz. the tangent and two from the centre to meet it,) since the line from the centre to the point of contact subtends a right angle, it is greater than the other line from the centre to the tangent, but it is to a part of it, (def. 15, b. 1,) which is absurd. Therefore it is evident that no other line from the centre, but this to the point of contact, is at right angles to the tangent.