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

### From inside the book

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Page 222

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**cones**and cylinders KL , MN : the first**cone**has to the second , and the first cylinder to the second , the tripli- cate ratio of that which AC has to EG . For if the**cone**ABCDL have not to the**cone**EFGHN the triplicate ratio of that ... Page 223

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**cone**ABCDL has to X , the triplicate ratio of that which AC has to EG ; therefore , as the**cone**ABCDL is to X , so is the pyramid DQATBYCVL , to the pyramid HSEOFPGRN . But the same**cone**is greater than the pyramid contained in it ... Page 224

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**cone**is to the**cone**, so is the cylinder to the cylinder ; for ( XII . 10. ) every**cone**is a third part of the cylinder upon the same base , and of the same altitude . Therefore also the cylinder has to the cylinder , the triplicate ... Page 225

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**cone**ABG to the**cone**CDK , because ( XII . 10. ) the cylinders are triple of the**cones**. Therefore also GH is to KL , as the**cone**ABG to the**cone**CDK , and the cylinder EB to the cylinder FD . Wherefore**cones**, & c . A PROP . XV . THEOR ... Page 226

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**cones**and cylinders of the same altitude being to one another as their bases ; therefore ( V. A. ) the base BD is equal to the base FH ; and as BD is to FH , so is MN to KL . But let the altitudes KL , MN be unequal , and MN the greater ...### Other editions - View all

The First Six and the Eleventh and Twelfth Books of Euclid's Elements: With ... Euclid No preview available - 2016 |

### Common terms and phrases

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

### Popular passages

Page 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.

Page 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.

Page 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.

Page 4 - A rhombus is that which has all its sides equal, but its angles are not right angles.

Page 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.

Page 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...

Page 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.

Page 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.

Page 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.

Page 36 - PARALLELOGRAMS upon the same base, and between the same parallels, are equal to one another...