## 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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**propositions**of the fifth Book than those of any other of the Elements . In the second Book , also , some algebraic signs have been introduced , for the sake of representing more readily the addition and subtraction of the rectangles on ... Page 5

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**proposition**, requiring no formal demonstration to prove the truth of it ; but is received and assented to as soon as mentioned . Such as , the whole of any thing is greater than a part of it ; or , the whole is equal to all its parts ... Page 6

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**proposition**to be true , by proving that some absurdity would necessarily follow if the**proposition**advanced were false . This is sometimes called Reductio ad Absurdum ; because it shows the absurdity and falsehood of all ... Page 7

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**proposition**and the Book in which it has been announced or de- monstrated . The expression ( 15. 1. ) denotes the fifteenth**proposition**, first book , and so on . In like manner , ( 3. Ax . ) designates the third axiom ; ( 2. Post ... Page 11

... to one another . 11. " Two straight lines which intersect one another , cannot be both pa- " rallel to the same straight line . "

... to one another . 11. " Two straight lines which intersect one another , cannot be both pa- " rallel to the same straight line . "

**PROPOSITION**I. PROBLEM . To describe an equilateral triangle upon OF GEOMETRY . BOOK I. 11.### 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 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 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 third 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...