## The Elements of Euclid |

### Inni boken

Resultat 1-5 av 31

Side 12

the angle BAC is

coincide with the point F , because the straight line AC is

point B coincides with the point E ; wherefore the base BC shall coincide with the

base EF ...

the angle BAC is

**equal**to the angle EDF ; wherefore also the point C shallcoincide with the point F , because the straight line AC is

**equal**to DF : but thepoint B coincides with the point E ; wherefore the base BC shall coincide with the

base EF ...

Side 25

to

also the third angle of the one to tire third angle of the other . Let ABC , DEF be

two triangles which have the angles ABC , BCA

viz ...

to

**equal**angles in each ; then shall the other sides be**equal**, each to each ; andalso the third angle of the one to tire third angle of the other . Let ABC , DEF be

two triangles which have the angles ABC , BCA

**equal**to the angles DEF , EFD ,viz ...

Side 57

others , that which is nearer to the line which passes through the centre is always

greater than one more remote ; and from the same point there can be drawn only

two straight lines that are

others , that which is nearer to the line which passes through the centre is always

greater than one more remote ; and from the same point there can be drawn only

two straight lines that are

**equal**to one another , one upon each side of the ... Side 71

angle BAC to the angle EDF . If the angle BGC be

manifest ( 20 . 3 . ) that the angle BAC is also

**equal**circumferences BC , EF ; the angle BGC is**equal**to the angle EHF , and theangle BAC to the angle EDF . If the angle BGC be

**equal**to the angle EHF , it ismanifest ( 20 . 3 . ) that the angle BAC is also

**equal**to EDF : but , if not , one of ... Side 87

in the alternate segment of the circle ; to each of these add the angle CDA :

therefore the whole angle BDA is

exterior angle BCD is

BDA is ...

in the alternate segment of the circle ; to each of these add the angle CDA :

therefore the whole angle BDA is

**equal**to the two angles CDA , DAC ; but theexterior angle BCD is

**equal**( 32 . 1 . ) to the angles CDA , DAC ; therefore alsoBDA is ...

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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.