## Elements of Geometry: Containing the First Six Books of Euclid: With a Supplement on the Quadrature of the Circle and the Geometry of Solids ...J. Eastburn & Company, 1819 - 317 pages |

### From inside the book

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**Ratio**is a mutual relation of two magnitudes , of the same kind , to one another , in respect of quantity . IV . Magnitudes are said to be of the same kind , when the less can be mul- tiplied so as to exceed the greater ; and it is only ... Page 109

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**ratio**than the third magnitude has to the fourth ; and , on the contrary , the third is said to have to the fourth a less**ratio**than the first has to the second . 1 . VIII . 1 When there is any number of magnitudes greater than two ... Page 110

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**ratio**, which is compounded of two equal**ratios**, is dupli- " cate of either of these**ratios**. " XII . If four magnitudes are continual proportionals , the**ratio**of the first to the fourth is said to be triplicate of the**ratio**of the ... Page 113

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**ratio**to the second which the third has to the fourth , and if any equimultiples whatever be taken of the first and third , and any whatever of the second and fourth ; the multiple of the first shall have the same**ratio**to the ... Page 115

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**ratio**to the same magnitude ; and the same has the same**ratio**to equal magnitudes . Let A and B be equal magnitudes , and C any other ; A : C :: B : C. Let mA , mB , be any equimultiples of A and B ; and nC any multiple of C. Because A ...### Other editions - View all

### Common terms and phrases

ABC is equal ABCD altitude angle ABC angle ACB angle BAC arch AC base BC bisected centre circle ABC circumference common section cosine cylinder demonstrated diameter draw equal and similar equal angles equiangular equilateral equilateral polygon equimultiples Euclid exterior angle fore four right angles given straight line greater hypotenuse inscribed join less Let ABC Let the straight line BC magnitudes meet opposite angle parallel parallelogram perpendicular polygon prism produced proportionals proposition Q. E. D. COR Q. E. D. PROP radius ratio rectangle contained rectilineal figure remaining angle segment semicircle shewn side BC sine solid angle solid parallelepipeds spherical angle spherical triangle SPHERICAL TRIGONOMETRY straight line AC THEOR third touches the circle triangle ABC triangle DEF wherefore

### Popular passages

Page 153 - If from the vertical angle of a triangle a straight line be drawn perpendicular to the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle...

Page 19 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.

Page 33 - THE greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it.

Page 292 - If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles...

Page 308 - 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 36 - IF two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other ; the base of that which has the greater angle shall be greater than the base of the other.

Page 18 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle ; and the straight line which stands on the other is called a perpendicular to it.

Page 78 - To draw a straight line from a given point, either without or in the circumference, which shall touch a given circle. First, let A be a given point without the given circle BCD : it is required to draw a straight line from A which shall touch the circle.

Page 77 - THE straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle...

Page 39 - If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another ; and the exterior angle equal to the interior and opposite upon the same side ; and likewise the too interior angles upon the same side together equal to two right angles.