## 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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Side 207

L E M M A. I. * IF from the greater of two unequal magnitudes, there be taken

more than its

at length remain a magnitude less than the least of the proposed magnitudes.

L E M M A. I. * IF from the greater of two unequal magnitudes, there be taken

more than its

**half**, and from the remainder more than its**half**; and so on : there willat length remain a magnitude less than the least of the proposed magnitudes.

Side 218

If the cylinder be not triple of the cone, it must either be greater than the triple or

less than it. First, let it be greater than the triple; and describe the square ABCD in

the circle; this square is greater than the

If the cylinder be not triple of the cone, it must either be greater than the triple or

less than it. First, let it be greater than the triple; and describe the square ABCD in

the circle; this square is greater than the

**half**of the circle AC, as was shown in ... Side 219

In the circle AC describe a square; this square is greater than the

: and upon the square ABCD form a pyramid having the same vertex with the

cone; this pyramid is greater than the

In the circle AC describe a square; this square is greater than the

**half**of the circle: and upon the square ABCD form a pyramid having the same vertex with the

cone; this pyramid is greater than the

**half**of the cone; because, as was before ... Side 241

are equal: therefore ECD is

because AD, AC are equal, AH common, and the contained angles equal, the

angles at H (I. 4.) are equal, and are therefore right angles. Then, in the triangles

AEF, ...

are equal: therefore ECD is

**half**of BED. Again, in the triangles AHD, AHC,because AD, AC are equal, AH common, and the contained angles equal, the

angles at H (I. 4.) are equal, and are therefore right angles. Then, in the triangles

AEF, ...

Side 306

the remainder is A + B. Take the

a + b : a - b : : tang (A+B): tan 3 (A–B). This analogy gives

and B; and” by adding this and 3 (A + B) together, A, the greater angle, ...

the remainder is A + B. Take the

**half**of this, and then, by the third proposition, asa + b : a - b : : tang (A+B): tan 3 (A–B). This analogy gives

**half**the difference of Aand B; and” by adding this and 3 (A + B) together, A, the greater angle, ...

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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...