## The Elements of Euclid |

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Side 297

found to the sum of the two magnitudes by which the given ratio is exhibited, one

of them, and the

found to the sum of the two magnitudes by which the given ratio is exhibited, one

of them, and the

**given magnitude**; each of the parts is given.” Let the**given****magnitude**AB be divided into the parts AC, CB. which have a given ratio to one ... 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 AD be the

has a given ratio to BC; the ratio of DC to DB is given (7. dat.); make the ratio of

AD to DE the same with this ratio; therefore the ratio of AD to DE is given: and AD

is ...

Let AD be the

**given magnitude**; and because DB, the excess of AB above AD,has a given ratio to BC; the ratio of DC to DB is given (7. dat.); make the ratio of

AD to DE the same with this ratio; therefore the ratio of AD to DE is given: and AD

is ...

Side 304

EB above a

other case is demonstrated. PROP. XX. 16. IF to one of two magnitudes which

have a given ratio to one another, a

EB above a

**given magnitude**EG, has a given ratio to FD. In the same manner theother 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 from the ... Side 307

ratio to the second, the same excess shall have a given ratio to the first; as is

evident from thd 9th dat. PROP. XXV. 17. If there be three magnitudes, the excess

of the first whereof above a

...

ratio to the second, the same excess shall have a given ratio to the first; as is

evident from thd 9th dat. PROP. XXV. 17. If there be three magnitudes, the excess

of the first whereof above a

**given magnitude**has a given ratio to the second; and...

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The Elements of Euclid: Viz. The First Six Books, Together With the Eleventh ... Robert Simson Ingen forhåndsvisning tilgjengelig - 2017 |

The Elements of Euclid: Viz. The First Six Books, Together With the Eleventh ... Robert Simson Ingen forhåndsvisning tilgjengelig - 2017 |

### Vanlige uttrykk og setninger

altitude angle ABC angle BAC base BC BC is equal BC is given bisected centre circle ABCD circumference cone cylinder demonstrated described diameter draw drawn equal angles equiangular equimultiples Euclid excess fore given angle given in magnitude given in position given in species given magnitude given point given ratio given straight line gles gnomon join less Let ABC meet multiple parallel parallelogram AC perpendicular point F polygon prism proportionals proposition pyramid Q. E. D. PROP radius ratio of AE rectangle CB rectangle contained rectilineal figure remaining angle right angles segment sides BA similar sine solid angle solid parallelopiped square of AC straight line AB straight line BC tangent THEOR third three plane angles triangle ABC triplicate ratio vertex wherefore

### Populære avsnitt

Side 45 - Ir a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.

Side 41 - If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part.

Side 54 - Ir any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle. Let ABC be a circle, and A, B any two points in the circumference ; the straight line drawn from A to B shall fall within the circle.

Side 18 - ABD, the less to the greater, which is impossible ; therefore BE is not in the same straight line with BC.

Side 10 - From a given point to draw a straight line equal to a given straight line. Let A be the given point, and BC the given straight line: it is required to draw from the point A a straight line equal to BC.

Side 8 - Let it be granted that a straight line may be drawn from any one point to any other point.

Side 256 - 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 129 - 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 23 - At a given point in a given straight line, to make a rectilineal angle equal to a given rectilineal angle. Let AB be the given straight line, and A...

Side 20 - ANY two angles of a triangle are together less than two right angles.