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

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Side 6

Things which are

Things which are

**doubles**of the same , or of equals , are equal to one another . 7. Things which are halves of the same , or of equals , are equal to one another . I 8. Magnitudes which exactly coincide with one another , are equal . 9. Side 35

This will be shown by drawing diagonals subtending the equal angles ; as ( I. 4. ) the triangles thus formed are equal ; and , by this proposition , the parallelograms are respectively

This will be shown by drawing diagonals subtending the equal angles ; as ( I. 4. ) the triangles thus formed are equal ; and , by this proposition , the parallelograms are respectively

**double**of ... Side 36

opposite to the base BC , be terminated in the same point D ; each of the parallelograms is

opposite to the base BC , be terminated in the same point D ; each of the parallelograms is

**double**( I. 34. ) of the triangle BDC ; and they are therefore equal ( I. ax . 6. ) to one another . But if the sides AD , EF ( figs . 2. and 3. ) ... Side 39

the triangles ABC , EBC are equal , because they are upon the same base BC , and between the same parallels BC , AE . But ( 1. 34. ) the parallelogram ABCD is

the triangles ABC , EBC are equal , because they are upon the same base BC , and between the same parallels BC , AE . But ( 1. 34. ) the parallelogram ABCD is

**double**of the triangle ABC , because the diagonal AC bisects it : wherefore ... Side 44

Now the parallelogram BL is

Now the parallelogram BL is

**double**... and between the same parallels BD , AL ; and the square GB is**double**of the triangle FBC , because these also are upon the same base FB , and between ... But the**doubles**of equals are equal ( I.ax.### Hva folk mener - Skriv en omtale

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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 evidently extremities figure fore four fourth given given circle given straight line greater half Hence inscribed join less magnitudes manner means meet method multiple opposite parallel 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 solid angles 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...