## Elements of Geometry: Containing the First Six Books of Euclid, with a Supplement on the Quadrature of the Circle, and the Geometry of Solids. To which are Added, Elements of Plane and Sperical Trigonometry |

### From inside the book

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**ratio**of one quantity to another , is usually denoted by placing one of the two quantities over the other , in the form of a fraction : A B thus , signifies the**ratio**or quotient arising from the division of the quantity A by B. In fact ... Page 106

... greater contains the less a cer- tain number of times exactly . 3.

... greater contains the less a cer- tain number of times exactly . 3.

**Ratio**is a mutual relation of two magnitudes , of the same kind , to one another , in respect of quantity . 4. Magnitudes are said to be of the same kind ELEMENTS ... Page 107

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**ratio**to one another . 5. If there be four magnitudes , and if any equimultiples whatsoever be taken of the first ...**ratio**that the third has to the fourth . 6. Magnitudes are said to be proportionals , when the first has the same**ratio**... Page 108

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**ratio**which A has to D , then , for shortness ' sake , M is said to have to Na**ratio**compounded of the same**ratios**which compound the**ratio**of A to D ; that is , a**ratio**compounded of the**ratios**of E to F , G to H , and K to L. 11. If ... Page 111

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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 multiple ...### Other editions - View all

### Common terms and phrases

ABC is equal ABCD adjacent angles altitude angle ABC angle ACB angle BAC base BC bisected centre chord circle ABC circumference cosine cylinder demonstrated described diameter divided draw equal and similar equal angles equiangular equilateral equilateral polygon equimultiples Euclid exterior angle fore four right angles given rectilineal given straight line greater Hence hypotenuse inscribed join less Let ABC magnitudes meet multiple opposite angle parallel parallelogram parallelopiped perpendicular polygon prism PROB PROP proportional proposition quadrilateral radius ratio rectangle contained rectilineal figure remaining angle right angled triangle SCHOLIUM segment semicircle shewn side BC sine solid angle solid parallelopiped spherical angle spherical triangle square straight line BC THEOR touches the circle triangle ABC triangle DEF wherefore

### Popular passages

Page 95 - 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. either the sides adjacent to the equal...

Page 68 - THE angles in the same segment of a circle are equal to one another...

Page 23 - Straight lines which are parallel to the same straight line are parallel to one another. Triangles and Rectilinear Figures. The sum of the angles of a triangle is equal to two right angles.

Page 74 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

Page 78 - IF from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it.

Page 9 - UPON the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise those which are terminated in the other extremity.

Page 75 - If a straight line touch a circle, and from the point of contact a...

Page 18 - AT a given point in a given straight line, to make a rectilineal angle equal to a given rectilineal angle. Let AB be the given straight line, and A...

Page 134 - EQUIANGULAR parallelograms have to one another the ratio which is compounded of the ratios of their sides.

Page 136 - AB is (7. 5.) to AD, as AE to AG ; and DC to CB, as GF to FE; and also CD to DA, as FG to GA ; therefore the sides of the parallelograms ABCD, AEFG about the equal angles are proportionals; and they are therefore similar to one another (1.