## 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 Spherical Trigonometry |

### From inside the book

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**multiple**of B by m . When the num- " ber is intended to multiply two or more magnitudes that follow , it is " written thus , m ( A + B ) , which signifies the sum of A and B taken m " times ; m ( A - B ) is m times the excess of A above ... Page 107

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**multiple**of the first is greater than the**multiple**of the second , equal to it , or less , the**multiple**of the third is also greater than the**multiple**of the fourth , equal to it , or less ; then the first of the magnitudes is said to ... Page 109

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**multiples**, are equal to one another . " 3. A**multiple**of a greater magnitude is greater than the same**multiple**of a less . 4. That magnitude of which a**multiple**is greater than the same multi- ple of another , is greater than that ... Page 110

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**multiple**of D + E + F by m . PROP . II . THEOR . r If to a**multiple**of a magnitude by any number , a**multiple**of the same mag- nitude by any number be added , the sum will be the same**multiple**of that magnitude that the sum of the two ... Page 111

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**multiple**of the second , that the**multiple**of the third has to the**multiple**of the fourth . : Let A B C D , and let m and n be any two numbers ; mA : nB :: mC : nD . Take of mA and mC equimultiples by any number p , and of nB and nD ...### 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 Let the straight 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 spherical angle spherical triangle square straight line BC THEOR touches the circle triangle ABC triangle DEF wherefore

### Popular passages

Page 51 - If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.

Page 12 - AB; but things which are equal to the same are equal to one another...

Page 80 - 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 288 - 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 35 - Parallelograms upon the same base and between the same parallels, are equal to one another.

Page 81 - If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles made by this line with the line which touches the circle, shall be equal to the angles in the alternate segments of the circle.

Page 52 - 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 in the point C ; the squares of AB, BC are equal to twice the rectangle AB, BC, together with the square of AC.

Page 127 - 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 23 - If, from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than, the other two sides of the triangle, but shall contain a greater angle.

Page 19 - The angles which one straight line makes with another upon one side of it, are either two right angles, or are together equal to two right angles. Let the straight line AB make with CD, upon one side of it the angles CBA, ABD ; these are either two right angles, or are together equal to two right angles. For, if the angle CBA be equal to ABD, each of them is a right angle (Def.