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

### From inside the book

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Page 41

Let the parallelograms

Let the parallelograms

**ABCD**, EBCF be constituted upon the same base Bc , and between the same parallels AF , BC : I say , the parallelogram**ABCD**is equal ... Page 42

is equal to the parallelogram

is equal to the parallelogram

**ABCD**; for it has the fame base BC , and is constituted between the fame parallels , BC , A H. By the like way of reasoning ... Page 45

cuts it into halves : wherefore the parallelogram

cuts it into halves : wherefore the parallelogram

**ABCD**в с will be double to the triangle E B C. If therefore a parallelogram and a triangle have the same ... Page 48

Let the given right - lined figure be

Let the given right - lined figure be

**ABCD**, and the given right - lined angle be E : It is required to constitute a parallelogram equal to the given right ... Page 49

And because the triangle A B D is equal to the parallelogram MF , and the triangle A B C to the parallelogram GM , the whole right - line figure

And because the triangle A B D is equal to the parallelogram MF , and the triangle A B C to the parallelogram GM , the whole right - line figure

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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...