## The Elements of Euclid |

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Resultat 1-5 av 7

Side 47

Ir a straight line be bisected , and produced to any point , the square of the whole

line thus produced and the square of the part of it produced , are together double

of the square of

Ir a straight line be bisected , and produced to any point , the square of the whole

line thus produced and the square of the part of it produced , are together double

of the square of

**half**the line bisected , and of the square of the line made up of ... Side 200

If from the greater of two unequal magnitudes , there be taken more than its

and from the remainder more than its

remain a magnitude less than the least of the proposed magnitudes . * Let AB

and C ...

If from the greater of two unequal magnitudes , there be taken more than its

**half**,and from the remainder more than its

**half**, and so on : there shall at lengthremain a magnitude less than the least of the proposed magnitudes . * Let AB

and C ...

Side 216

In the circle EFGH describe the square EFGH , therefore this square is greater

than the

altitude with the cone ; this pyramid is greater than

square ...

In the circle EFGH describe the square EFGH , therefore this square is greater

than the

**half**of the circle : upon the square EFGH erect a pyramid of the samealtitude with the cone ; this pyramid is greater than

**half**of the cone . For , if asquare ...

Side 225

the circle EFGH : then , if the circumference BAD be bisected , and the

again bisected , and so on , there must at length remain a circumference less (

Lemma . ) than AD : let this be A LD ; and from the point L draw LM 29 BOOK XII .

the circle EFGH : then , if the circumference BAD be bisected , and the

**half**of it beagain bisected , and so on , there must at length remain a circumference less (

Lemma . ) than AD : let this be A LD ; and from the point L draw LM 29 BOOK XII .

Side 391

Euclid Robert Simson. the square of the radius is to the square of the tangent of

plane triangle , the base is to the sum of the sides , as the difference of the sides ...

Euclid Robert Simson. the square of the radius is to the square of the tangent of

**half**the angle BAC opposite to the base . Q . E . D . PROP . VII . FIG . 12 . 13 . In aplane triangle , the base is to the sum of the sides , as the difference of the sides ...

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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.