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BOOK II.

DEFINITIONS.

I.

EVERY right-angled parallelogram is called a rectangle, and is said to be contained by any two of the straight lines which contain one of the right angles.

II.

In every parallelogram, any of the parallelograms about a diameter, together with the two complements, is called a gnomon.

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If there be two straight lines, one of which is divided into any number of parts; the rectangle contained by the two straight lines, is equal to the rectangles contained by the undivided line, and the several parts of the divided line.

PROP. II. THEOREM.

If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts, are together equal to the square of the whole line.

PROP. III. THEOREM.

If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part.

PROP. IV. THEOREM.

If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts.

COR. From the demonstration, it is manifest, that the parallelograms about the diameter of a square are likewise squares.

PROP. V. THEOREM.

If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.

COR. From this proposition it is manifest, that the difference of the squares of two unequal lines, AC, CD, is equal to the rectangle contained by their sum and difference.

PROP. VI. THEOREM.

If a straight line be bisected, and produced to any point; the rectangle contained by the whole line thus produced, and the part of it produced, together with the square of half the line bisected, is equal to the square of the straight line which is made up of the half and the part produced.

PROP. VII. THEOREM.

If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part.

PROP. VIII. THEOREM.

If a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line, which is made up of the whole and that part.

PROP. IX. THEOREM.

If a straight line be divided into two equal, and also into two unequal parts; the squares of the two unequal parts are together double of the square of half the line, and of the square of the line between the points of

section.

PROP. X. THEOREM.

If a straight line be bisected, and produced to any point, the square of the whole line thus produced, and the square of the part of it produced, are together double of the square of half the line bisected, and of the square of the line made up of the half and the part produced.

PROP. XI. PROBLEM.

To divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts, shall be equal to the square of the other part.

PROP. XII. THEOREM.

In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle is greater than the squares of the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted without the triangle between the perpendicular and the obtuse angle.

PROP. XIII. THEOREM.

In every triangle, the square of the side subtending any of the acute angles, is less than the squares of the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the acute angle and the perpendicular let fall upon it from the opposite angle.

PROP. XIV. PROBLEM.

To describe a square that shall be equal to a given rectilineal figure.

BOOK III.

DEFINITIONS.

I.

EQUAL circles are those of which the diameters are equal, or from the centres of which the straight lines to the circumferences are equal.

II.

A straight line is said to touch a circle, when it meets the circle, and being produced does not cut it.

III.

Circles are said to touch one another, which meet, but do not cut one another.

IV.

Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal.

V.

And the straight line on which the greater perpendicular falls, is said to be farther from the centre.

VI.

A segment of a circle, is the figure contained by a straight line and the circumference which it cuts off.

VII.

The angle of a segment, is that which is contained by the straight line and the circumference.

VIII.

An angle in a segment, is the angle contained by two straight lines drawn from any point in the circumference of the segment, to the extremities of the straight line which is the base of the segment.

IX.

An angle is said to insist or stand upon the circumference intercepted between the straight lines that contain the angle.

X.

A sector of a circle, is the figure contained by two straight lines drawn from the centre, and the circumference between them.

XI.

Similar segments of circles are those in which the angles are equal, or which contain equal angles.

P. D. E.

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PROPOSITIONS.

PROP. I. PROBLEM.

To find the centre of a given circle.

COR. From this it is manifest, that if in a circle a straight line bisect another at right angles, the centre of the circle is in the line which bisects the other.

PROP. II. THEOREM.

If any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle.

PROP. III. THEOREM.

If a straight line drawn through the centre of a circle, bisect a straight line in it which does not pass through the centre, it shall cut it at right angles and conversely, if it cut it at right angles, it shall bisect it.

PROP. IV. THEOREM.

If in a circle two straight lines cut one another, which do not both pass through the centre, they do not bisect each other.

PROP. V. THEOREM.

If two circles cut one another, they shall not have the same centre.

PROP. VI. THEOREM.

If one circle touch another internally, they shall not have the same

centre.

PROP. VII. THEOREM.

If any point be taken in the diameter of a circle, which is not the centre, of all the straight lines which can be drawn from it to the circumference, the greatest is that in which the centre is, and the other part of that diameter is the least; and, of any others, that which is nearer to the line which passes through the centre, is always greater than one more remote and from the same point there can be drawn only two equal straight lines to the circumference, one upon each side of the diameter.

PROP. VIII. THEOREM.

If any point be taken without a circle, and straight lines be draw from it to the circumference, whereof one passes through the centre; of those which fall upon the concave circumference, the greatest is that which passes through the centre; and of the rest, that which is nearer to that through the centre is always greater than the more remote: but of those which fall upon the convex circumference, the least is that between the point without the circle and the diameter; and of the rest, that which is nearer to the least is always less than the more remote: and only two equal straight lines can be drawn from the same point to the circumference, one upon each side of the line which passes through the centre.

PROP. IX. THEOREM.

If a point be taken within a circle, from which there fall more than two equal straight lines to the circumference, that point is the centre of the circle.

PROP. X. THEOREM.

One circumference of a circle cannot cut another in more than two points.

PROP. XI. THEOREM.

If one circle touch another internally in any point, the straight line which joins their centres being produced shall pass through that point of

contact.

PROP. XII. THEOREM.

If two circles touch each other externally in any point, the straight line which joins their centres, shall pass through that point of contact.

PROP. XIII. THEOREM.

One circle cannot touch another in more points than one, whether it touches it on the inside or outside.

PROP. XIV. THEOREM.

Equal straight lines in a circle are equally distant from the centre; and those which are equally distant from the centre, are equal to one another. PROP. XV. THEOREM.

The diameter is the greatest straight line in a circle; and, of all others, that which is nearer to the centre, is always greater than one more remote: and the greater is nearer to the centre than the less.

PROP. XVI. THEOREM

The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn from the extremity between that straight line and the circumference, so as not to cut the circle; or, which is the same thing, no straight line can make so great an acute angle with the diameter at its extremity, or so small an angle with the straight line which is at right angles to it, as not to cut the circle.

COR. From this it is manifest, that the straight line which is drawn at right angles to the diameter of a circle from the extremity of it, touches the circle; and that it touches it only in one point, because, if it did meet the circle in two, it would fall within it. Also, it is evident, that there can be but one straight line which touches the circle in the same point.'

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PROP. XVII. PROBLEM.

To draw a straight line from a given point, either without or in the circumference, which shall touch a given circle.

PROP. XVIII. THEOREM.

If a straight line touch a circle, the straight line drawn from the centre to the point of contact, shall be perpendicular to the line touching the circle.

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