## The First Six and the Eleventh and Twelfth Books of Euclid's Elements: With Notes and Illustrations, and an Appendix in Five Books |

### Inni boken

Side 39

the same parallels BF , AG ; but the

DEF ; therefore also the triangle GEF is equal to the triangle DEF , the less to the

greater , which is impossible . Therefore AG is not parallel to BF ; and in the same

...

the same parallels BF , AG ; but the

**triangle ABC**is equal ( hyp . ) to the triangleDEF ; therefore also the triangle GEF is equal to the triangle DEF , the less to the

greater , which is impossible . Therefore AG is not parallel to BF ; and in the same

...

Side 137

P GB с D к L of BC , the same multiple is the triangle AHC of ABC . ... viz . , the

two bases BC , CD , and the two

the

...

P GB с D к L of BC , the same multiple is the triangle AHC of ABC . ... viz . , the

two bases BC , CD , and the two

**triangles ABC**, ACD ; and of the base BC , andthe

**triangle ABC**, the first and third , any like multiples whatever have been taken...

Side 155

as BC to BG , so is the

has to ABG the duplicate ratio of that which BC has to EF . But the triangle ABG is

equal to DEF ; wherefore also the

as BC to BG , so is the

**triangle ABC**to ABG . Therefore ( V. 11. ) the**triangle ABC**has to ABG the duplicate ratio of that which BC has to EF . But the triangle ABG is

equal to DEF ; wherefore also the

**triangle ABC**has to DEF the duplicate ratio of ... Side 212

XL is parallel to AB , and the

, DEF is similar to RVF . Now , because BC is double of CX , and EF of FV ,

therefore BC : CX :: EF : FV ; and upon BC , CX are described the similar and

similarly ...

XL is parallel to AB , and the

**triangle ABC**is similar to LXC . For the same reason, DEF is similar to RVF . Now , because BC is double of CX , and EF of FV ,

therefore BC : CX :: EF : FV ; and upon BC , CX are described the similar and

similarly ...

Side 269

But ED is two - thirds of AD , and therefore GI is two - thirds of ABC ; wherefore

the space between the circumferences of GI ... D C G If on BC , the hypotenuse of

a right - angled

side ...

But ED is two - thirds of AD , and therefore GI is two - thirds of ABC ; wherefore

the space between the circumferences of GI ... D C G If on BC , the hypotenuse of

a right - angled

**triangle ABC**, a semicircle BFAGC be described on the sameside ...

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The First Six and the Eleventh and Twelfth Books of Euclid's Elements: With ... Euclid Ingen forhåndsvisning tilgjengelig - 2016 |

### Vanlige uttrykk og setninger

ABCD altitude base bisected called centre chord circle circumference coincide common cone consequently const construction contained continual cylinder demonstrated describe diagonal diameter difference divided double draw equal equal angles extremities figure fore four fourth given given circle given point given straight line greater half Hence inscribed join less magnitudes manner means meet method multiple opposite parallel parallelepiped parallelogram pass perpendicular plane polygon prism PROB produced proof PROP proportional proposition proved pyramid radius ratio reason rectangle remaining respectively right angles Schol segments semicircle shown sides similar square straight line taken THEOR third touching triangle triangle ABC twice vertical wherefore whole

### Populære avsnitt

Side 94 - 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.

Side 53 - 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 143 - 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 4 - A rhombus is that which has all its sides equal, but its angles are not right angles.

Side 57 - To divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts may be equal to the square on the other part.

Side 138 - IF a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those produced, proportionally; and if the sides, or the sides produced, be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle...

Side 43 - In any right-angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle.

Side 32 - 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 40 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

Side 36 - PARALLELOGRAMS upon the same base, and between the same parallels, are equal to one another...