## The Elements of Euclid |

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Side 252

EM not before

construction to one , which , A without doubt , is Euclid ' s , in which nothing is

required but to add a

be ...

EM not before

**given**the method of doing . For this reason , we have changed theconstruction to one , which , A without doubt , is Euclid ' s , in which nothing is

required but to add a

**magnitude**to itself a certain number of times ; and this is tobe ...

Side 300

with a

magnitude AB together with the

ratio to the magnitude CD ; the excess of CD above a

given ratio ...

with a

**given magnitude**has a given ratio to the first magnitude . * Let themagnitude AB together with the

**given magnitude**BE , that is , AE , have a givenratio to the magnitude CD ; the excess of CD above a

**given magnitude**has agiven ratio ...

Side 301

If the excess of a magnitude , above a

another magnitude ; the excess of both together above a

have to that other a given ratio : and if the excess of two magnitudes together

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**shallhave to that other a given ratio : and if the excess of two magnitudes together

above a ...

Side 302

Let the excess of the magnitude AB above a

to the magnitude BC : the excess of AB above a

ratio to AC . Let AD be the

...

Let the excess of the magnitude AB above a

**given magnitude**have a given ratioto the magnitude BC : the excess of AB above a

**given magnitude**has a givenratio to AC . Let AD be the

**given magnitude**; and because DB , the excess of AB...

Side 304

In the EB above a

the other case is demonstrated . PROP . XX . 16 . If to one of two magnitudes

which have a given ratio to one another , a

the ...

In the EB above a

**given magnitude**EG , has a given ratio to FD . same mannerthe other case is demonstrated . PROP . XX . 16 . If to one of two magnitudes

which have a given ratio to one another , a

**given magnitude**be added , and fromthe ...

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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.