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

### From inside the book

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Page 117

... the others that which is nearer to the least will always be

more remote ; and only two equal right lines can fall upon the circumference of

ibe circle , the one on one side the least line , and the other on the other side of it

.

... the others that which is nearer to the least will always be

**less**than that which ismore remote ; and only two equal right lines can fall upon the circumference of

ibe circle , the one on one side the least line , and the other on the other side of it

.

Page 118

D K

circle A B C , and join ME , MF , MC , MK , ML , MH . Now because A M is equal to

EM , let MD , which is common , be added ; and then A D is equal to EM , MD .

D K

**less**than Dl , and DL**less**than DH . For [ by 1. 3. ] find the centre m of thecircle A B C , and join ME , MF , MC , MK , ML , MH . Now because A M is equal to

EM , let MD , which is common , be added ; and then A D is equal to EM , MD .

Page 202

In like manner a regular polygon of thirty two sides infcribed in the circle , will

differ

ftill be nearer to the circle in which it is inscribed , than one of thirty two sides ;

and so ...

In like manner a regular polygon of thirty two sides infcribed in the circle , will

differ

**less**from the circle than one of sixteen fides ; and one of fixty four fides willftill be nearer to the circle in which it is inscribed , than one of thirty two sides ;

and so ...

Page 205

EU C L I D's E L E M E N T S. BOOK V. DEFINITION S. I. A Part is a magnitude of

a magnitude , a

multiple is a greater ( magnitude ] of a

lefs ...

EU C L I D's E L E M E N T S. BOOK V. DEFINITION S. I. A Part is a magnitude of

a magnitude , a

**less**of a greater , when the**less**measures the greater . 2. Amultiple is a greater ( magnitude ] of a

**less**, when the greater is measured by thelefs ...

Page 208

If equal , equal ; if

proposed magnitudes , and any equimultiples of the first and third be taken , as

also any other equimultiples of the second and fourth ; and the equimultiple of the

...

If equal , equal ; if

**less**,**less**. Wherefore , on the contrary , when there are fourproposed magnitudes , and any equimultiples of the first and third be taken , as

also any other equimultiples of the second and fourth ; and the equimultiple of 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.

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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.

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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...