## The Elements of Euclid |

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Resultat 1-5 av 7

Side 199

If there be two triangular

a parallelogram , and the base of the other a triangle ; if the parallellogram be

double of the triangle , the

If there be two triangular

**prisms**of the same altitude , the base of one of which isa parallelogram , and the base of the other a triangle ; if the parallellogram be

double of the triangle , the

**prisms**shall be equal to one another . Let the**prisms**... Side 205

of the triangle GFC : but when there are two

one has a parallelogram for its base ... to one another ; therefore the

the parallelogram EBFG for its base , and the straight line KH opposite to it , is ...

of the triangle GFC : but when there are two

**prisms**of the same altitude , of whichone has a parallelogram for its base ... to one another ; therefore the

**prism**havingthe parallelogram EBFG for its base , and the straight line KH opposite to it , is ...

Side 207

parts in the points N , Y by the same planes : therefore the

RVFSTY are of the same altitude ; and therefore as the base LXC to the base

RVF ; that is , as the triangle ABC to the triangle DEF , so ( Cor . 32 . 11 . ) is the

parts in the points N , Y by the same planes : therefore the

**prisms**LXCOMN ,RVFSTY are of the same altitude ; and therefore as the base LXC to the base

RVF ; that is , as the triangle ABC to the triangle DEF , so ( Cor . 32 . 11 . ) is the

**prism**... Side 209

the solid Q is greater than the

which is impossible . Therefore the base ABC is not to the base DEF , as the

pyramid ABCG to any solid which is less than the pyramid DEFH . In the same

manner ...

the solid Q is greater than the

**prisms**in the pyramid DEFH . But it is also less ,which is impossible . Therefore the base ABC is not to the base DEF , as the

pyramid ABCG to any solid which is less than the pyramid DEFH . In the same

manner ...

Side 211

From this it is manifest , that every pyramid is the third part of a

the same base , and is of an equal altitude with it ; for if the base of the

any other figure than a triangle , it may be divided into

From this it is manifest , that every pyramid is the third part of a

**prism**which hasthe same base , and is of an equal altitude with it ; for if the base of the

**prism**beany other figure than a triangle , it may be divided into

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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.