## Euclid's Elements of Geometry: The First Six, the Eleventh and Twelfth Books |

### From inside the book

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Page 110

A sector of a circle is the figure comprehended between two right lines drawn

from the

is between them . II . Similar fegments of circles are those which receive equal ...

A sector of a circle is the figure comprehended between two right lines drawn

from the

**centre**of the circle and that part of the circumference of the circle whichis between them . II . Similar fegments of circles are those which receive equal ...

Page 113

drawn thro ' the

angles . For [ by 1. 3. ] find the

E A , E B. Then because a F is equal to FB , and FE is common , two sides A F ,

EF ...

drawn thro ' the

**centre**into two equal parts : I say , it will cut the same at rightangles . For [ by 1. 3. ] find the

**centre**of the circle A BC , which let be e , and joinE A , E B. Then because a F is equal to FB , and FE is common , two sides A F ,

EF ...

Page 114

Let there be a circle A B C D , and in it let two right lines AC , B D not drawn

through the

one another into two equal parts . For , if possible , let them bisect each other so

that ...

Let there be a circle A B C D , and in it let two right lines AC , B D not drawn

through the

**centre**intersect one ano , ther in the point E : I say , these will not cutone another into two equal parts . For , if possible , let them bisect each other so

that ...

Page 115

Therefore the point E is not the

to be demonstrated . Therefore if two circles cut one another , they will not both

have the same

Therefore the point E is not the

**centre**of both the circles A BC , CDG . Which wasto be demonstrated . Therefore if two circles cut one another , they will not both

have the same

**centre**. PROP . VI . THEOR . If two circles touch one another ... Page 120

the angles AED , BED are both right angles ; wherefore since GK bilecis A B , it

will cut the same at right angles ; and because if one right line in H a circle

bisects a right line in it at right angles , the

) ...

the angles AED , BED are both right angles ; wherefore since GK bilecis A B , it

will cut the same at right angles ; and because if one right line in H a circle

bisects a right line in it at right angles , the

**centre**of the circle will fall [ by cor . I. 3.) ...

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### Common terms and phrases

A B C ABCD added alſo altitude baſe becauſe centre circle circumference common cone cylinder definition demonſtrated deſcribed diameter difference divided double draw drawn equal equal angles equiangular equimultiples Euclid exceeds fall fame fides figure firſt folid fore four fourth given right line greater half inſcribed join leſs magnitudes manner meet multiple oppoſite parallel parallelogram perpendicular plane polygon priſms PROP proportional propoſition proved pyramid ratio rectangle remaining angle right angles right line A B right lined figure ſame ſay ſecond ſegment ſhall ſides ſimilar ſince ſolid ſome ſphere ſquare ſtand ſum taken THEOR theſe third thoſe thro touch triangle triangle ABC twice vertex Wherefore whole whoſe baſe

### Popular passages

Page 247 - 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 30 - 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 248 - But it was proved that the angle AGB is equal to the angle at F ; therefore the angle at F is greater than a right angle : But by the hypothesis, it is less than a right angle ; which is absurd.

Page 18 - When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

Page 32 - Let the straight line EF, which falls upon the two straight lines AB, CD, make the alternate angles AEF, EFD equal to one another; AB is parallel to CD.

Page 56 - Therefore all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

Page 391 - KL: but the cylinder CM is equal to the cylinder EB, and the axis LN to the axis GH; therefore as the cylinder EB to...

Page 110 - If any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle.

Page 130 - When you have proved that the three angles of every triangle are equal to two right angles...

Page 183 - FK : in the same manner it may be demonstrated, that FL, FM, FG are each of them equal to FH, or FK : therefore the five straight lines FG, FH, FK, FL, FM are equal to one another : wherefore the circle described from the centre F, at the distance of...