## The Elements of Euclid |

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Side 57

Let

which is not the centre ; let the centre be E ; of all the straight lines FB , FC , FG , &

c . that can be drawn from F to the circumference , FA is the greatest , and FD , the

...

Let

**ABCD**be a**circle**, and AD its diameter , in which let any point E be takenwhich is not the centre ; let the centre be E ; of all the straight lines FB , FC , FG , &

c . that can be drawn from F to the circumference , FA is the greatest , and FD , the

...

Side 84

And it is manifest , that when the centre of the circle falls within the triangle , each

of its angles is less than a right angle ... It is also rectangular ; for the straight line

BD , being the diameter of the

And it is manifest , that when the centre of the circle falls within the triangle , each

of its angles is less than a right angle ... It is also rectangular ; for the straight line

BD , being the diameter of the

**circle ABCD**, BAD is a semicircle ; wherefore the ... Side 87

in the alternate segment of the

therefore the whole angle BDA is equal to the ... the whole

whole EDCB : and the angle AED stands on the circumference

angle ...

in the alternate segment of the

**circle**; to each of these add the angle CDA :therefore the whole angle BDA is equal to the ... the whole

**ABCD**is equal to thewhole EDCB : and the angle AED stands on the circumference

**ABCD**, and theangle ...

Side 202

CIRCLES are to one another as the squares of their diameters . * Let ABCD ,

EFGH be two circles , and BD , FH their diameters : as the square of BD to the

square of FH , so is the

square ...

CIRCLES are to one another as the squares of their diameters . * Let ABCD ,

EFGH be two circles , and BD , FH their diameters : as the square of BD to the

square of FH , so is the

**circle ABCD**, to the circle EFGH . For , if it be not so , thesquare ...

Side 203

excess of the circle EFGH above the space S : because , by the preceding lemma

, if from the greater of two unequal ... the polygon AXBOCPDR to the polygon

EKFLGMHN : but the

excess of the circle EFGH above the space S : because , by the preceding lemma

, if from the greater of two unequal ... the polygon AXBOCPDR to the polygon

EKFLGMHN : but the

**circle ABCD**is greater than the polygon contained in it ...### Hva folk mener - Skriv en omtale

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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.