## The Elements of Euclid |

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Resultat 1-5 av 6

Side 26

Let the straight line EF , which falls upon the two straight lines AB , CD make the

alternate angles AEF , EFD equal to one another ; AB is parallel to CD . For , if it

be not parallel , AB and CD being

Let the straight line EF , which falls upon the two straight lines AB , CD make the

alternate angles AEF , EFD equal to one another ; AB is parallel to CD . For , if it

be not parallel , AB and CD being

**produced**shall meet either towards B , D , or ... Side 43

If a straight line be bisected , and

by the whole line thus

square of half the line bisected , is equal to the square of the straight line which is

...

If a straight line be bisected , and

**produced**to any point ; the rectangle containedby the whole line thus

**produced**, and the part of it**produced**, together with thesquare of half the line bisected , is equal to the square of the straight line which is

...

Side 47

Ir a straight line be bisected , and

line thus

of the square of half the line bisected , and of the square of the line made up of ...

Ir a straight line be bisected , and

**produced**to any point , the square of the wholeline thus

**produced**and the square of the part of it**produced**, are together doubleof the square of half the line bisected , and of the square of the line made up of ...

Side 166

For , if it be not , EF , GH shall meet , if

first , let them be

since EFK is in the plane AB , every point in EFK is in that plane ; and K is a ...

For , if it be not , EF , GH shall meet , if

**produced**, either on the side of FH , or EG ;first , let them be

**produced**on the side of FH , and meet in the point K ; therefore ,since EFK is in the plane AB , every point in EFK is in that plane ; and K is a ...

Side 185

... to the solid CF :

parallel to DA ; and let HB , OD

LR ; therefore the solid AE , of which the base is the parallelogram LE , and AK

the ...

... to the solid CF :

**produce**DL , TS , until they meet in A , and from B draw BHparallel to DA ; and let HB , OD

**produced**meet in Q , and complete the solids AE ,LR ; therefore the solid AE , of which the base is the parallelogram LE , and AK

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.