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

### From inside the book

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Page 6

gles , those right lines , being infinitely produced , do

angles are less than two right angles ; 12. Two right lines do not comprehend a

space . t All these axioms are so evident , when the words by which they are ...

gles , those right lines , being infinitely produced , do

**meet**on that fide where theangles are less than two right angles ; 12. Two right lines do not comprehend a

space . t All these axioms are so evident , when the words by which they are ...

Page 35

I say , that o G D the lines A B , CD may for ever be thus prolonged , yet never

fhall they

B D being cut in half in k the line K B shall be equal to B L , and K D to DL .

Wherefore ...

I say , that o G D the lines A B , CD may for ever be thus prolonged , yet never

fhall they

**meet**together . For , if posible , let them**meet**in the point l ; thereforeB D being cut in half in k the line K B shall be equal to B L , and K D to DL .

Wherefore ...

Page 175

I. ] draw D F ; E F from the points Dg E at right angles to A B , AC ; these will either

E B ACB . с B

triangle A B C. First let them

FC ...

I. ] draw D F ; E F from the points Dg E at right angles to A B , AC ; these will either

E B ACB . с B

**meet**within the triangle ABC , in the side B C , or without thetriangle A B C. First let them

**meet**within the triangle in the point F , and join B F ,FC ...

Page 176

Wherefore if the given triangle ABC be an acute angled one , DE , EF will

within the triangle ; but if it has a right angle B A C , they will

If B A C be an obtuse angle , they will

Wherefore if the given triangle ABC be an acute angled one , DE , EF will

**meet**within the triangle ; but if it has a right angle B A C , they will

**meet**in the side bc :If B A C be an obtuse angle , they will

**meet**without the triangle A B C. PROP . VI . Page 328

For if they be not parallel , EF , GH being produced will either

or towards E , G. Let them first be produced and

Then because E Fk is in the plane A B , all the points taken in EFK will be in the ...

For if they be not parallel , EF , GH being produced will either

**meet**towards F , H ,or towards E , G. Let them first be produced and

**meet**towards F , H , viz , at K :Then because E Fk is in the plane A B , all the points taken in EFK will be in the ...

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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.

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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...