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If not in each circle draw from the centre right lines to the extremities of the given line.

Then because the circles are their radii are te one another and the given lines are,.. the angles at the centres are, (prop. 8, b. 1,) and.. the arches on which they stand are, (prop. 26, b. 3,) take away those =arches from the circles and the remaining arches

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PROP. 29, THEOR.

In equal circles, the right lines which subtend equal arches, are equal.

If the equal arches be semicircles, the proposition is evident.

But if not, draw from the centre in each, right lines to the extremities of the given lines.

Then because the arches subtended by the given lines are , the angles at the centres standing on those arches are =, but in the triangles to which those angles belong, the sides about one angle are respectively to those about the other, as being radii of circles, .. the lines subtending them are,.. &c.

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Schol. Whatever has been demonstrated in the ceding propositions of equal circles, is also true of the same circle.

PROP. 30. PROB.

To bisect a given, arch.

Connect its extremities, bisect the connecting line, and draw from the point of bisection a line at right angles to it to meet the circumference: it bisects the arch,

For, connect the point in which it meets the circumference, with the extremities of the bisected line.

Then in the triangles formed by those connecting lines, the half lines and perpendicular; a half line and perpendicular in one are respectively to the other half line and perpendicular in the other, and the contained angles are being right,.. the lines opposite to them or the connecting lines are,.. the arches which they subtend are .. &c.

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PROP. 31, THEOR.

In a circle, the angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is an acute angle, and the angle in a segment less than a semicircle is obtuse.

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PART 1. The angle in a semicircle is a right angle. Draw a diameter connecting the extremities of the sides about the angle, and join the vertex of it with the centre.

Then this connecting line being to each part of the diameter at each side of the centre, (by prop. 5, b. 1,) the sum of the angles contained by it and each side of the given angle, i. e. the angle in the given semicircle, is = to the sum of the angles contained by the diameter and sides about the given angle, .. (by cor. 1, prop. 82, 1,) this angle is right,.. &c.

PART 2. The angle in a segment greater than a semicircle is acute.

Draw through the extremity, (remote from the given angle,) of either side about it, a diameter connecting the said extremities, and draw through the extremity of that line, not conterminous with the diameter, a right line connecting the adjacent extremity of the diameter with it, this connecting line forms with the diameter an angle to the given angle since they are in the same segment, but the angle contained by the subtense of the given angle and this connecting line is a right angle, (part 1). the angle contained by the diameter and the line from its extremity to the extremity of the side not conterminous with it, is less than a right angle, and.. the given angle is less than a right angle.

PART 3. The angle in a segment less than a semi

circle is obtuse.

Take in the opposite circumference any point and connect it with the adjacent extremities of the given sides.

Then, in the quadrilateral figure thus formed, the given angle and angle at the assumed point are to two right -angles, (prop. 22, b. 3,) but the angle at the assumed point is an acute angle, (part 2, præc.).. the given angle is

obtuse.

Cor. 1. Hence can be derived a method of drawing through the extremity of any right line a perpendicular to it; take any point without it, and with this point as a centre describe a circle passing through the extremity and cutting the given line in any point; draw through this point a diameter, the right line connecting the extremity with the extremity of the diameter remote from the given line is at right angles to the given line, because it is the angle in a semicircle.

Cor, 2. Hence also can be drawn a tangent to a circle from a given point without it: draw a right line from the given point to the centre of the circle, bisect it, and from the point of bisection as centre, describe a circle to pass through the given point: the right line drawn from either intersection of this circle with the given circle is a tangent; for it is perpendicular to the radius drawn to the point where it meets the circle, because the angle in a semicircle is a right angle.

PROP. 32, THEOR.

If a right line be a tangent to a circle, and from the point of contact a right line be drawn cutting the circle, the angle made by this line with the tangent, is equal to the angle in the alternate segment of the circle.

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If the secant should pass through the centre, it is evident that the angles are, for each of them is a right angle, (prop. 18 & 31, b. 3.)

But if not, draw through the point of contact a perpen dicular to the tangent, connect its extremity with that of the secant, viz. those extremities remote from the point of contact.

Then in the triangle thus formed, the sum of the angles adjacent to the perpendicular is to a right angle, (for the perpendicular passes through the centre,) but one of them is the complement of the alternate angle contained by the secant and tangent, .. the other is to it, 1. e. the angle contained by the secant and tangent is to that in the alternate segment, (this is the angle in the segment greater than a semicircle.)

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Also the angle in the segment less than a semicircle is to the alternate angle contained by the secant and tangent.

For it, with that in the other seg. is to two right angles, and also the sum of those at the point of contact is to two right angles, but one of those in the foregoing case has been proved to the angle in the greater seg. the other is to that in the lesser seg.

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PROP. 33, PROB.

On a given right line, to describe a segment of a circle that shall contain an angle equal to a given rectilineal angle.

If the given angle be right, bisect the given line, and describe a circle (with half of it as radius,) the circle is divided by the given line into two semicircles, .. each of them contains an angle to the given right angle, (prop. 31, b. 3.)

But if the given angle be acute or obtuse, make with the given line at either extremity of it an angle to the given angle, draw through the vertex of this angle a line at right angles to the drawn line, and draw from the other extremity of the given line, a line making an angle with it to the angle which the perpendicular makes with the given line, the point where this line meets the perpendicular is the centre of the required circle.

For this line is to that part of the perpendicular between it and the right angle, since the angles contained by them and the given line are,.. the circle described from this as a centré, and either of those lines from it to the given lines as radius, will pass through the extremities of the given line, and the line making the given angle with it is a tangent to this circle, since it is at right angles to its radius, .. &c.

Schol. In the same manner a circle can be described which shall contain an angle to a given angle.

PROP. 34. PROB.

To cut off from a given circle a segment which shall contain an angle equal to a given angle.

Draw a tangent to the given circle, and draw through the point of contact a secant making an angle to the given angle.

This secant evidently cuts off the required segment.

PROP. 35, THEOR.

If two right lines within a circle cut one another, the rectangle under the segments of one of them, is equal to the rectangle under the segments of the other.

1. If the given right lines pass through the centre, they are bisected in the point of intersection, .. the rectangles under the segments are the 'rs of their halves, and are...

2. Let one of the given lines pass through the centre, and the other not; join the centre and the extremities of that not passing through it,

Then the rectangle under the segments of that not passing through the centre is to the difference of the O'rs of the intercept (between the centre and intersection of given lines) and either of the connecting lines, (cor. prop. 6, b. 2,) i. e. to the difference between the O'rs of this intercept and half the line passing through centre, but the rect. under the segments of the line passing through the centre is also to the dif. between thers of its half and intercept, (prop. 5, b. 2,). the rectangles are

3. Let neither of them pass through the centre; draw through their intersection a diameter. Then (by part 2, præc) the rect. under the segments of each of the given lines, is to the rectangle under the segments of this diameter, .. &c.

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