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

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Resultat 1-5 av 6

Side 11

the angle BAC is equal to the angle EDF;

coincide with the point F, because (hyp.) AC is equal to DF. But the point B

coincides with E;

because, the point ...

the angle BAC is equal to the angle EDF;

**wherefore**also the point C shallcoincide with the point F, because (hyp.) AC is equal to DF. But the point B

coincides with E;

**wherefore**the base BC shall coincide with the base EF;because, the point ...

Side 41

to two right angles; L * {*—is

two right angles; and therefore (I. ax. 12.) HB, FE will meet, if produced: let them

meet in K, and through K draw KL parallel to EA or FH, and produce HA, GB to

the ...

to two right angles; L * {*—is

**wherefore**the angles BHF, HFE H. A are less thantwo right angles; and therefore (I. ax. 12.) HB, FE will meet, if produced: let them

meet in K, and through K draw KL parallel to EA or FH, and produce HA, GB to

the ...

Side 116

B C E AG : DK:: GH : KL, and as HB : LE. And (W. 12.) as one of the antecedents

to its consequent, so are all the antecedents together to all the consequents

together;

Therefore ...

B C E AG : DK:: GH : KL, and as HB : LE. And (W. 12.) as one of the antecedents

to its consequent, so are all the antecedents together to all the consequents

together;

**wherefore**as AG: DK: ; AB : DE: but AG is equal to C, and DK to F.Therefore ...

Side 143

the triangles ABC, GEF have their sides opposite to the equal angles

proportionals;

Therefore (V, 11.) DE : EF : : GE : EF; D whence, since DEand GE have the same

ratio /\ to EF, ...

the triangles ABC, GEF have their sides opposite to the equal angles

proportionals;

**wherefore**AB : BC : : GE : EF; but (hyp.) A B : BC : : DE : EF. ATherefore (V, 11.) DE : EF : : GE : EF; D whence, since DEand GE have the same

ratio /\ to EF, ...

Side 183

First, let them be produced towards FH, and meet in the point K. Then, since EFK

is in the plane AB, K is in A.B. For the same reason, K is also in CD:

the planes AB, CD I' th produced meet one another; but they do not meet, since ...

First, let them be produced towards FH, and meet in the point K. Then, since EFK

is in the plane AB, K is in A.B. For the same reason, K is also in CD:

**wherefore**the planes AB, CD I' th produced meet one another; but they do not meet, since ...

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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 angle ABC angle BAC angle equal BC is equal bisected centre chord circle ABC circumference cone const contained cylinder describe a circle diagonal diameter divided draw equal angles equal to AC equiangular equilateral Euclid exterior angle fore fourth given circle given point given ratio given straight line greater half Hence hypotenuse inscribed join less Let ABC magnitudes manner multiple opposite parallel parallelepiped parallelogram perpendicular polygon polyhedron prism PROB produced PROP proportional proposition pyramid radius rectangle rectilineal figure right angles Schol segments semicircle sides similar similar triangles solid angles square of AC straight lines drawn tangent THEOR third triangle ABC triplicate ratio vertex vertical angle wherefore

### 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...