## The Elements of Euclid |

### Inni boken

Resultat 1-5 av 14

Side 57

than the base FC : for the same

GF , FE , are greater ( 20 . 1 . ) than EG , and EG is equal to ED ; GF , FE are

greater than ED : take away the DH common part FE , and the remainder GF is ...

than the base FC : for the same

**reason**CF , is greater than GF : again , becauseGF , FE , are greater ( 20 . 1 . ) than EG , and EG is equal to ED ; GF , FE are

greater than ED : take away the DH common part FE , and the remainder GF is ...

Side 84

to the base AD ; and for the same

or AD ; therefore the quadrilateral figure ABCD is equilateral . It is also

rectangular ; for the straight line BD , being the diameter of the circle ABCD , BAD

is a ...

to the base AD ; and for the same

**reason**, BC , CD are each of them equal to BA ,or AD ; therefore the quadrilateral figure ABCD is equilateral . It is also

rectangular ; for the straight line BD , being the diameter of the circle ABCD , BAD

is a ...

Side 88

to the angle AED : for the same

equal to the angle BAE , or AED : therefore the pentagon ABCDE is equiangular ;

and it has been shown that it is equilateral . Wherefore , in the given circle , an ...

to the angle AED : for the same

**reason**, each of the angles ABC , BCD , CDE isequal to the angle BAE , or AED : therefore the pentagon ABCDE is equiangular ;

and it has been shown that it is equilateral . Wherefore , in the given circle , an ...

Side 129

for the same

DE is equal to EG , and EF common , the two sides DE , EF are equal to the two

GE , EF , and the base DF is equal to the base GF : therefore the angle DEF is ...

for the same

**reason**, DF is equal to FG : and because in the triangles DEF , GEF ,DE is equal to EG , and EF common , the two sides DE , EF are equal to the two

GE , EF , and the base DF is equal to the base GF : therefore the angle DEF is ...

Side 138

... the whole angle AGH is equal to the whole CFE : for the same

angle ABH is equal to the angle CDE ; also the angle at A is equal to the angle at

C , and the angle GHB to FED ; therefore the rectilineal figure ABHG is

equiangular ...

... the whole angle AGH is equal to the whole CFE : for the same

**reason**, theangle ABH is equal to the angle CDE ; also the angle at A is equal to the angle at

C , and the angle GHB to FED ; therefore the rectilineal figure ABHG is

equiangular ...

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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.