## The Elements of Euclid |

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Resultat 1-5 av 6

Side 215

Therefore the rest of the cylinder , that is , the prism of which the base is the

polygon AEBFCGDH , and of which the altitude is the same with that of the

cylinder , is greater than the triple of the

12 . ) ...

Therefore the rest of the cylinder , that is , the prism of which the base is the

polygon AEBFCGDH , and of which the altitude is the same with that of the

cylinder , is greater than the triple of the

**cone**: but this prism is triple ( 1 . Cor . 7 .12 . ) ...

Side 216

part of the cylinder . Let these be the segments upon AE , EB , BF , FC , CG , GD ,

DH , HA . Therefore the rest of the

is ...

**cone**, which together shall be less than the excess of the**cone**above the thirdpart of the cylinder . Let these be the segments upon AE , EB , BF , FC , CG , GD ,

DH , HA . Therefore the rest of the

**cone**, that is , the pyramid , of which the baseis ...

Side 218

the base is the other polygon , and its vertex N : therefore , as the

solid X , so is the pyramid of which the base is the polygon ATBYCVDQ , and

vertex L , to the pyramid the base of which is the polygon EOFPGRHS , and

vertex N ...

the base is the other polygon , and its vertex N : therefore , as the

**cone**AL to thesolid X , so is the pyramid of which the base is the polygon ATBYCVDQ , and

vertex L , to the pyramid the base of which is the polygon EOFPGRHS , and

vertex N ...

Side 221

But by the hypothesis , the

L , has to the solid X , the triplicate ratio of that which AC has to EG : therefore as

the

But by the hypothesis , the

**cone**of which the base is the circle ABCD , and vertexL , has to the solid X , the triplicate ratio of that which AC has to EG : therefore as

the

**cone**of which the base is the circle ABCD , and vertex L , is to the solid X ... Side 222

ABCDL , so is the

the

therefore the

...

ABCDL , so is the

**cone**EFGHN to some solid , which must be less ( 14 . 5 . ) thanthe

**cone**ABCDL , because the solid Z . is greater than the**cone**EFGHN :therefore the

**cone**EFGHN has to a solid which is less than the**cone**ABCDL , the...

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added altitude angle ABC angle BAC base BC is given centre circle circle ABCD circumference common cone contained cylinder definition demonstrated described diameter divided double draw drawn equal equal angles equiangular equimultiples Euclid excess fore four fourth given angle given in magnitude given in position given in species given magnitude given ratio given straight line gles greater Greek half join less likewise magnitude manner meet multiple Note opposite parallel parallelogram pass perpendicular plane prism produced PROP proportionals proposition pyramid Q. E. D. PROP reason rectangle rectangle contained remaining right angles segment shown sides similar sine solid sphere square square of BC taken THEOR third triangle ABC wherefore whole

### Populære avsnitt

Side 36 - If a straight line be divided into any two parts, the squares of the whole line and of one of the parts are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Let the straight line AB be divided into any two parts at the point C : the squares of AB, BC shall be equal to twice the rectangle AB, BC, together with the square of AC.

Side 145 - Wherefore, in equal circles &c. QED PROPOSITION B. THEOREM If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.

Side 65 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

Side 248 - Again ; the mathematical postulate, that " things which are equal to the same are equal to one another," is similar to the form of the syllogism in logic, which unites things agreeing in the middle term.

Side 11 - If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Side 121 - 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.

Side 21 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

Side 14 - To construct a triangle of which the sides shall be equal to three given straight lines ; but any two whatever of these lines must be greater than the third (20.

Side 80 - 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...

Side 133 - ... rectilineal figures are to one another in the duplicate ratio of their homologous sides.