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

### From inside the book

Results 1-5 of 6

Page 36

Those right lines that are

another . Let each of the right lines A B , CD be

also , A B will be

because ...

Those right lines that are

**parallel**to the same right line , are**parallel**to oneanother . Let each of the right lines A B , CD be

**parallel**to the right line er : I sayalso , A B will be

**parallel**to C D. For let the right line G K fall upon them . Thenbecause ...

Page 39

is

two right lines which join two equal and

parts , are themselves both equal and

is

**parallel**to the right line B D. It has also been proved to be equal to it . Thereforetwo right lines which join two equal and

**parallel**right lines , towards the sameparts , are themselves both equal and

**parallel**. Which was to be demonstrated . Page 44

Equal triangles , constituted towards the same parts , upon the same base , are

between the same

is

...

Equal triangles , constituted towards the same parts , upon the same base , are

between the same

**parallels**. E Let the equal triangles ... For draw AD : I fay , thisis

**parallel**to Bc . For if not , through the point a [ by prop . 31. ) draw the right line...

Page 328

are

Therefore if two right lines touching one another be

touching one another , but not in the same plane ; those planes which pass thro ...

are

**parallel**: Therefore the planes paffing thro ' A B , BC ; DE , EP are par . allel .Therefore if two right lines touching one another be

**parallel**to two right linestouching one another , but not in the same plane ; those planes which pass thro ...

Page 341

For because the two

common sections [ by 16. 11. ) are

Again , becaufe the two

common ...

For because the two

**parallel**planes BG , CE are cut by the plane A C , theircommon sections [ by 16. 11. ) are

**parallel**: Therefore A B is**parallel**to DC .Again , becaufe the two

**parallel**planes B F , A E are cut by the plane AC , theircommon ...

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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...