## The Elements of Euclid |

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Side 249

... where mention is made of a

any thing in the proposition relating to a ... Analogy is the similitude of

the same kind , and can serve for no purpose in mathematics , but only to give ...

... where mention is made of a

**given**angle , though there neither is , nor can beany thing in the proposition relating to a ... Analogy is the similitude of

**ratios**, is ofthe same kind , and can serve for no purpose in mathematics , but only to give ...

Side 257

... of an uncertain proposition : upon this account , we have

demonstration of this proposition instead of that in ... I say the

to the fourth D , is either equal to , or greater , or less than the

the ...

... of an uncertain proposition : upon this account , we have

**given**a legitimatedemonstration of this proposition instead of that in ... I say the

**ratio**of the third Cto the fourth D , is either equal to , or greater , or less than the

**ratio**of the first A tothe ...

Side 265

pression by which the

at the same time , is shown that there are ... and a more simple demonstration is

...

pression by which the

**ratio**of the first A to the third C is signified , and by which ,at the same time , is shown that there are ... and a more simple demonstration is

**given**from the fourth prop . of this book , the same which is in the translation from...

Side 301

If the excess of a magnitude , above a given magnitude , has a

another magnitude ; the excess of both together above a given magnitude shall

have to that other a

above a ...

If the excess of a magnitude , above a given magnitude , has a

**given ratio**toanother magnitude ; the excess of both together above a given magnitude shall

have to that other a

**given ratio**: and if the excess of two magnitudes togetherabove a ...

Side 302

Let the excess of the magnitude AB above a given magnitude have a

to the magnitude BC : the excess of AB above a given magnitude has a

...

Let the excess of the magnitude AB above a given magnitude have a

**given ratio**to the magnitude BC : the excess of AB above a given magnitude has a

**given****ratio**to AC . Let AD be the given magnitude ; and because DB , the excess of AB...

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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.