## 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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**equimultiples**whatsoever be taken of the first and third , and any**equimultiples**whatsoever of the se- cond and fourth , and if , according as the multiple of the first is greater than the multiple of the second , equal to it , or less ... Page 109

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**EQUIMULTIPLES**of the same , or of equal magnitudes , are equal to one another . 2. Those magnitudes of which the same , or equal magnitudes , are**equi- multiples**, are equal to one another . 3. A multiple of a greater magnitude is ... Page 110

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**equimultiple**of D , E , and F , it would be shown that A + B + C was the same multiple of D + E + F . COR . Hence , if m be any number , mD + mE + mF = m ( D + E + F ) . For mD , mE , and mF are**multiples**of D , E , and F by m ... Page 111

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**equimultiples**by any number p , and of nB and nD**equimultiples**by any number q . Then the**equimultiples**of mA , and mC by p , are**equimultiples**also of A and C , for they contain A and C as oft as there are units in pm ( 3. 5. ) , and ... Page 112

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**equimultiples**of A and C ; nB and nD any**equi- multiples**of B and D. Then , because A : B :: C : D , if mA be less than nB , mC will be less than nD ( def . 5. 5. ) , that is , if nB be greater than mA , nD will be greater than mC . For ...### 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 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 149 - IF an angle of a triangle be bisected by a straight line, which likewise cuts the base ; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of...

Page 12 - To draw a straight line through a given point parallel to a given straight line. Let A be the given point, and BC the given straight line.

Page 9 - 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 52 - If a straight line be bisected, and produced to any point, the rectangle contained by the whole line thus produced, and the part of it produced, together with the square on half the line bisected, is equal to the square on the straight line which is made up of the half and the part produced.

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 296 - 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 50 - If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part.

Page 15 - 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 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...