## The Elements of Euclid |

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Resultat 1-5 av 15

Side 20

ABC be any triangle ; any two of its angles together are less than two right angles

. Produce BC to D ; and because ACD is the exterior angle of the triangle ABC ...

**THEOR**. Any two angles of a triangle are together less than two right angles . LetABC be any triangle ; any two of its angles together are less than two right angles

. Produce BC to D ; and because ACD is the exterior angle of the triangle ABC ...

Side 32

PROP . XXXVI .

the same parallels , are equal to one another . Let ABCD , EFGH be pa . A D E H

rallelograms upon equal bases BC , FG , and between the same ! parallels AH ...

PROP . XXXVI .

**THEOR**. PARALLELOGRAMS upon equal bases , and betweenthe same parallels , are equal to one another . Let ABCD , EFGH be pa . A D E H

rallelograms upon equal bases BC , FG , and between the same ! parallels AH ...

Side 56

V .

uaw C Let the two circles ABC , CDG cut one another in the points B , C ; they

have not the same centre . For , if it be possible , let E be their centre : join EC ,

and ...

V .

**THEOR**. If two circles cut one another , they shall not have the same centre .uaw C Let the two circles ABC , CDG cut one another in the points B , C ; they

have not the same centre . For , if it be possible , let E be their centre : join EC ,

and ...

Side 57

to the circumference , whereof one passes through the centre , of those which fall

upon the concave circumference , the greatest is that which passes through the ...

**THEOR**. If any point be taken without a circle , and straight lines be drawn from itto the circumference , whereof one passes through the centre , of those which fall

upon the concave circumference , the greatest is that which passes through the ...

Side 60

their centres being produced shall pass through the point of contact . Let the two

circles ABC , ADE touch each other internally in the point A , and let F be the

centre ...

**THEOR**. If two circles touch each other internally , the straight line which joinstheir centres being produced shall pass through the point of contact . Let the two

circles ABC , ADE touch each other internally in the point A , and let F be the

centre ...

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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.