## The Elements of Euclid |

### Inni boken

Resultat 1-5 av 14

Side 89

angle FLC : and because KC is equal to CL , KL is double of KC : in the same

manner , it may be

KC , as was demonstrated , and that KL is double of KC , and HK double of BK ,

HK ...

angle FLC : and because KC is equal to CL , KL is double of KC : in the same

manner , it may be

**shown**that HK is double of BK : and because BK is equal toKC , as was demonstrated , and that KL is double of KC , and HK double of BK ,

HK ...

Side 105

3 .

than D ; therefore the whole EG is greater than K and D together ; but , K together

with D , is equal to L ; therefore EG is greater than L ; but FG is not greater ...

3 .

**shown**, that FG was E not less than K , and , by the construction , EF is greaterthan D ; therefore the whole EG is greater than K and D together ; but , K together

with D , is equal to L ; therefore EG is greater than L ; but FG is not greater ...

Side 113

but it was

GK is Bi the same multiple of AE , that LN is of CF ; that is , GK , LN are equimulE

ft tiples of AE , CF : and because KO , NP are equimultiples of BE , DF , if G CL ...

but it was

**shown**that LM is the same multiple of CD , that GK is of AE ; thereforeGK is Bi the same multiple of AE , that LN is of CF ; that is , GK , LN are equimulE

ft tiples of AE , CF : and because KO , NP are equimultiples of BE , DF , if G CL ...

Side 238

... KBCL , does nevertheless remain : therefore the line BC has no breadth : and

because the line BC is a superficies , and that a superficies has no thickness , as

was

... KBCL , does nevertheless remain : therefore the line BC has no breadth : and

because the line BC is a superficies , and that a superficies has no thickness , as

was

**shown**; therefore a line has neither breadth nor thickness , but only length . Side 241

262 of the second edition of his Elements of Geometry , printed anno 1760 ,

observes in his notes , that it ought to have been

below the line EG . This probably Euclid omitted , as it is very easy to perceive ,

that DG ...

262 of the second edition of his Elements of Geometry , printed anno 1760 ,

observes in his notes , that it ought to have been

**shown**that the point F fallsbelow the line EG . This probably Euclid omitted , as it is very easy to perceive ,

that DG ...

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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.