## The Elements of Euclid |

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Resultat 1-5 av 7

Side 216

cone , which together shall be less than the

part 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 base

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 base

is ...

Side 274

But if B and C together be less than A ; then , because it is required that B , C , D

together be greater than A , from each of these taking away B , C , the remaining

one D must be greater than the

But if B and C together be less than A ; then , because it is required that B , C , D

together be greater than A , from each of these taking away B , C , the remaining

one D must be greater than the

**excess**of A above B and C ; take therefore any ... Side 301

If the

another magnitude ; the

have to that other a given ratio : and if the

above a ...

If the

**excess**of a magnitude , above a given magnitude , has a given ratio toanother magnitude ; the

**excess**of both together above a given magnitude shallhave to that other a given ratio : and if the

**excess**of two magnitudes togetherabove a ...

Side 306

than AG : - and AB , AG are given ; therefore the remainder BG is given : and

because C F D as AE to CF , so is AG to CD , and so - is ( 19 . 5 . ) EG to FD ; the

ratio of EG to FE is given : and GB is given ; therefore EG , the

above a ...

than AG : - and AB , AG are given ; therefore the remainder BG is given : and

because C F D as AE to CF , so is AG to CD , and so - is ( 19 . 5 . ) EG to FD ; the

ratio of EG to FE is given : and GB is given ; therefore EG , the

**excess**of EBabove a ...

Side 307

ratio to the second , the same

evident from thd 9th dat . PROP . XXV . 17 . If there be three magnitudes , the

second ...

ratio to the second , the same

**excess**shall have a given ratio to the first ; as isevident from thd 9th dat . PROP . XXV . 17 . If there be three magnitudes , the

**excess**of the first whereof above a given magnitude has a given ratio to thesecond ...

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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.