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

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Page 112

Therefore the right line

, nor does it fall in the circumference . Consequently it must necessarily fall within

the circle . If therefore any two points be taken in the circumference of a circle ...

Therefore the right line

**drawn**from the point A to B does not fall without the circle, nor does it fall in the circumference . Consequently it must necessarily fall within

the circle . If therefore any two points be taken in the circumference of a circle ...

Page 113

angles . For [ by 1. 3. ] find the centre of the circle A BC , which let be e , and join

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 join

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

EF ...

Page 129

I say , the right line AB is

because E is the centre of the circles B C D , AFG ; E A will be equal to E F , and

E D equal to EB : Wherefore the two sides A E , E B are equal to F the two fides

F E ...

I say , the right line AB is

**drawn**from the point A to touch the circle ABC . Forbecause E is the centre of the circles B C D , AFG ; E A will be equal to E F , and

E D equal to EB : Wherefore the two sides A E , E B are equal to F the two fides

F E ...

Page 322

ED will be perpendicular to the plane

angles to all right lines which touch it , and are in that same plane . But dc is in

the plane

ED will be perpendicular to the plane

**drawn**thro ' BD , DA ; and will be at rightangles to all right lines which touch it , and are in that same plane . But dc is in

the plane

**drawn**thro ' BA , AD , because AB , BD [ by 2. 11. ) are in the plane**drawn**... Page 324

)

to BC . Then because B C is at right angles to AD , DE ; [ by 4 11. ) Be will be at ...

**draw**de in the given plane from the point d'y perpendicular to Bc ; and ( by 12 . 1.)

**draw**A F from the point A , perpendicular to DE ; and thro's D**draw**G H parallelto BC . Then because B C is at right angles to AD , DE ; [ by 4 11. ) Be will be at ...

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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...