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

### From inside the book

Results 1-5 of 5

Page 6

gles ,

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 188

And the part A B of the circumference , being the fifth part of the whole , will be

three of

equal parts [ by 30. 3. ) Bisect Bc in E : Then will BE , E c be each one fifteenth

part of ...

And the part A B of the circumference , being the fifth part of the whole , will be

three of

**those**equal parts : B and so the remaining part BC E is two of**those**equal parts [ by 30. 3. ) Bisect Bc in E : Then will BE , E c be each one fifteenth

part of ...

Page 198

Divide the quarter Dc of the circumference into the same number of equal parts

as the polygon to be described has sides . Then if the polygon has five sides , let

c e be one of

Divide the quarter Dc of the circumference into the same number of equal parts

as the polygon to be described has sides . Then if the polygon has five sides , let

c e be one of

**those**parts . If it has seven sides , let c E be three of**those**parts ; if ... Page 249

...

be demonstrated . ... less than a right angle :

and bave

...

**those**angles equal which are opposite to the homologous fides , which was tobe demonstrated . ... less than a right angle :

**those**triangles will be equiangular ,and bave

**those**angles equal about which are the proportional fides Let the two ... Page 429

The following definition therefore of fimilar segments of Circles , viz . that they are

convey a much clearer notion of fimilar segments than that other of Euclids , to

The following definition therefore of fimilar segments of Circles , viz . that they are

**those**whose bases are in the same proportion to their diameters , seems toconvey a much clearer notion of fimilar segments than that other of Euclids , to

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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...