## The Elements of Euclid |

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Resultat 1-5 av 17

Side 25

to equal angles in each ; then shall the other sides be equal , each to each ; and

also the

two triangles which have the angles ABC , BCA equal to the angles DEF , EFD ,

viz ...

to equal angles in each ; then shall the other sides be equal , each to each ; and

also the

**third**angle of the one to tire**third**angle of the other . Let ABC , DEF betwo triangles which have the angles ABC , BCA equal to the angles DEF , EFD ,

viz ...

Side 57

than the

EF , that is , AF , is А greater than BF : again , because BE is equal B to CE , and

FE common to the triangles BEF , CEF , the two sides BE , EF are equal to the ...

than the

**third**, BE , EF are greater than BF ; but AE is equal to EB ; therefore AE ,EF , that is , AF , is А greater than BF : again , because BE is equal B to CE , and

FE common to the triangles BEF , CEF , the two sides BE , EF are equal to the ...

Side 87

... at the base double of the

PROP . To inscribe an equilateral and equiangular pentagon in a given circle .

Let ABCDE be the given circle ; it is required to inscribe an equilateral and

equiangular ...

... at the base double of the

**third**angle . Which was to be done . PROP . XI .PROP . To inscribe an equilateral and equiangular pentagon in a given circle .

Let ABCDE be the given circle ; it is required to inscribe an equilateral and

equiangular ...

Side 94

N . B . When four magnitudes are proportionals , it is usually expressed by saying

, the first is to the second , as the

equimultiples of four magnitudes ( taken as in the fifth definition ) the multiple of

the first is ...

N . B . When four magnitudes are proportionals , it is usually expressed by saying

, the first is to the second , as the

**third**to the fourth . ' VII . When of theequimultiples of four magnitudes ( taken as in the fifth definition ) the multiple of

the first is ...

Side 99

If the first of four magnitudes has the same ratio to the second which the

to the fourth , then any equimultiples whatever of the first and

same ratio to any equimultiples of the second and fourth , viz . “ the equimultiple ...

If the first of four magnitudes has the same ratio to the second which the

**third**hasto the fourth , then any equimultiples whatever of the first and

**third**shall have thesame ratio to any equimultiples of the second and fourth , viz . “ the equimultiple ...

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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.