## The Elements of Euclid: The Errors, by which Theon, Or Others, Have Long Ago Vitiated These Books, are Corrected and Some of Euclid's Demonstrations are Restored. Also, to this Second Edition is Added the Book of Euclid's Data. In Like Manner Corrected. viz. the first six books, together with the eleventh and twelfth |

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Page 31

THE opposite sides and angles of parallelograms are equal to one another , and

the

Parallelogram is a four sided figure of which ibe opposite fides are parallel . and

the ...

THE opposite sides and angles of parallelograms are equal to one another , and

the

**diameter**bisects them , that is , divides them into two equal parts . N. B. AParallelogram is a four sided figure of which ibe opposite fides are parallel . and

the ...

Page 34

Book I. rallelogram EBCA , because the

DBC is the half of the parallelogram DBCF , because the C. 34 , 1.

bisects it . but the halves of equal things are equal d ; d . 7. Ax . therefore the ...

Book I. rallelogram EBCA , because the

**diameter**AB bisects it ; and the triangleDBC is the half of the parallelogram DBCF , because the C. 34 , 1.

**diameter**DCbisects it . but the halves of equal things are equal d ; d . 7. Ax . therefore the ...

Page 65

THEOR . ff any point be taken in the

of all the straight lines which can be drawn from it to the circumference , the

greatest is that in which the center is , and the other part of that

least ...

THEOR . ff any point be taken in the

**diameter**of a circle , which is not the center ,of all the straight lines which can be drawn from it to the circumference , the

greatest is that in which the center is , and the other part of that

**diameter**is theleast ...

Page 175

ABCD and AEFG are about the fame

G parallelogram BD have its

the

ABCD and AEFG are about the fame

**diameter**. For if not , let , if possible , the А.G parallelogram BD have its

**diameter**AHC in a different straight line from K H AFthe

**diameter**of the parallelo E F gram EG , and let GF meet AHC in H ; and thro ... Page 450

In BC the

given point D be taken , and from D let a straight line DA be drawn to any point A

in the circumference , and let AE be drawn at right angles to DA , and from the ...

In BC the

**diameter**of the circle ABC given in position , or in BC produced , let thegiven point D be taken , and from D let a straight line DA be drawn to any point A

in the circumference , and let AE be drawn at right angles to DA , and from the ...

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### Common terms and phrases

added alſo altitude angle ABC angle BAC baſe BC is given becauſe Book Book XI caſe circle circle ABCD circumference common cone cylinder Definition demonſtrated deſcribed diameter divided double draw drawn equal equal angles equiangular equimultiples exceſs fame fides firſt folid fore four fourth given angle given in poſition given in ſpecies given magnitude given ratio given ſtraight line greater Greek half join leſs likewiſe magnitude manner meet muſt oppoſite parallel parallelogram perpendicular plane priſm produced PROP proportionals Propoſition pyramid reaſon rectangle remaining right angles ſame ſecond ſegment ſhall ſhewn ſides ſimilar ſolid ſquare ſquare of AC taken THEOR theſe third thro triangle ABC wherefore whole

### Popular passages

Page 5 - Let it be granted that a straight line may be drawn from any one point to any other point.

Page 163 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.

Page 48 - 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. Let the straight line AB be divided into any two parts in the point C; the squares of AB, BC are equal to twice the rectangle AB, BC, together with the square of AC.

Page 73 - 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...

Page 105 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides adjacent to the equal...

Page 3 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.

Page 167 - Similar triangles are to one another in the duplicate ratio of their homologous sides.

Page 54 - AB be the given straight line ; it is required to divide it 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.

Page 47 - 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.

Page 37 - To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.