## The Elements of Euclid |

### Inni boken

Resultat 1-5 av 13

Side 20

Let ABC be any triangle ; any two of its angles together are less than two right

angles . Produce BC to D ; and because ACD is the exterior angle of the triangle

ABC , ACD is greater ( 16 . 1 . ) than the interior and opposite

...

Let ABC be any triangle ; any two of its angles together are less than two right

angles . Produce BC to D ; and because ACD is the exterior angle of the triangle

ABC , ACD is greater ( 16 . 1 . ) than the interior and opposite

**angle ABC**; to each...

Side 89

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 ... is also double of the angle CBF ;

therefore the angle ABF is equal GAM to the angle CBF ; wherefore the

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 ... is also double of the angle CBF ;

therefore the angle ABF is equal GAM to the angle CBF ; wherefore the

**angle****ABC**is ... Side 129

BCA ; wherefore the remaining

32 . 1 . ) , and the triangle

consequently they have their sides opposite to the в equal

BCA ; wherefore the remaining

**angle**BAC is equal to the remaining**angle**EGF (32 . 1 . ) , and the triangle

**ABC**is therefore equiangular to the triangle GEF ; andconsequently they have their sides opposite to the в equal

**angles**proportionals ... Side 169

equal to AE , and AB is common , the two DA , AB are equal to the two EA , AB ,

and the

than two right

...

equal to AE , and AB is common , the two DA , AB are equal to the two EA , AB ,

and the

**angle**DAB is equal to the**angle**... BCA , ACD , CDA , ADB are greaterthan two right

**angles**: and because the three**angles**of each of the triangles**ABC**...

Side 193

to the plane BAC : and AB is drawn in the plane BAC at right angles to the

common section BK of the two planes ... to the base EF , and the remaining

angles to the remaining angles : the

DEF : and the ...

to the plane BAC : and AB is drawn in the plane BAC at right angles to the

common section BK of the two planes ... to the base EF , and the remaining

angles to the remaining angles : the

**angle ABC**is therefore equal to the angleDEF : and the ...

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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.