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

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Page 209

That those quantities that differ from one another by ma nitudes less than any

asignable

That any four proposed

numbers ...

That those quantities that differ from one another by ma nitudes less than any

asignable

**magnitudes**, may be taken for iquals , or represent one another . 2.That any four proposed

**magnitudes**may either be accurately expressed innumbers ...

Page 210

If three

the duplicate of the ratia of the first to the second 8 . 11. If four

continual ] proportionals , the ratio of the first to the fourth is said to be triplicate of

...

If three

**magnitudes**are proportionals , the ratio of the first to the third is said to bethe duplicate of the ratia of the first to the second 8 . 11. If four

**magnitudes**be [continual ] proportionals , the ratio of the first to the fourth is said to be triplicate of

...

Page 212

Ordinate proportion is , when it shall be as an antecedent to a consequent , so is

an antecedent to a consequent ; and as a consequent is to some other

perturbate ...

Ordinate proportion is , when it shall be as an antecedent to a consequent , so is

an antecedent to a consequent ; and as a consequent is to some other

**magnitude**, ro is a consequent to some other**magnitude**P. 20. Inordinate orperturbate ...

Page 218

THEO R. If two

some parts taken away from them be equimultiples of she same

then the remainders will either be equal to the fame

equimultiples of ...

THEO R. If two

**magnitudes**be equimulsiples of two other mag . nitudes , andsome parts taken away from them be equimultiples of she same

**magnitudes**;then the remainders will either be equal to the fame

**magnitudes**, orequimultiples of ...

Page 219

Equal

A , B have ...

Equal

**magnitudes**bave the same ratio to the same**magnitude**, and the same**magnitude**has the same ratio to equal**magnitudes*** . Let A , B be equal**magnitudes**, and c any**magnitude**whatsoever : I say , each of thele**magnitudes**A , B have ...

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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...