## The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh and Twelfth : the Errors by which Theon, Or Others, Have Long Ago Vitiated These Books are Corrected, and Some of Euclid's Demonstrations are Restored : Also, the Book of Euclid's Data, in Like Manner Corrected : to this Edition are Also Annexed, Elements of Plane and Spherical Trigonometry |

### From inside the book

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**parallelograms**are equal to one another , and the diameter bisects them , that is , divides them into two equal ...**parallelogram**, of which BC is a diameter ; the opposite sides and angles of the figure are equal to one another ... Page 38

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**parallelogram**ACDB into two equal parts . Q. E. D. PROP . XXXV . THEOR . THE**PARALLELOGRAMS**upon the same base , and between the same parallels , are equal to one another . * Let the**parallelograms**ABCD , EBCF be upon the same base BC ... Page 39

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**parallelogram**; and it is equal ( 35. 1. ) to ABCD , because it is upon the same base BC , and between the same parallels BC , AD : for the like reason , the**parallelogram**EFGH is equal to the same EBCH : therefore also the**parallelogram**... Page 40

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**parallelogram**EBCA , because the diameter AB bi- sects ( 34. 1. ) it ; and the triangle DBC is the half of the ...**parallelogram**; and they are equal ( 36. 1. ) to one another , because they are upon equal bases BC , EF , and ... Page 41

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**parallelogram**and triangle be upon the same base , and between the same . parallels ; the**parallelogram**shall be double of the triangle . Let the**parallelogram**ABCD and the triangle EBC be upon F the same base BC , and between the ...### Other editions - View all

The Elements of Euclid: Viz, the First Six Books, Together with the Eleventh ... Robert Simson,Euclid Euclid No preview available - 2018 |

### Common terms and phrases

AC is equal altitude angle ABC angle BAC base BC BC is given bisected centre circle ABCD circumference cone cylinder demonstrated described diameter draw drawn equal angles equiangular equimultiples Euclid ex æquali excess fore given angle given in magnitude given in position given in species given magnitude given ratio given straight line gnomon greater join less Let ABC meet multiple parallel parallelogram perpendicular point F polygon prism proportionals proposition pyramid Q. E. D. PROP radius ratio of AE rectangle CB rectangle contained rectilineal figure remaining angle right angles segment side BC similar solid angle solid parallelopipeds sphere spherical angle square of AC straight line AB straight line BC tangent THEOR third triangle ABC vertex wherefore

### Popular passages

Page 11 - Let it be granted that a straight line may be drawn from any one point to any other point.

Page 155 - IF a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those produced, proportionally; and if the sides, or the sides produced, be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle...

Page 329 - Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides.

Page 20 - To draw a straight line at right angles to a given straight line, from a given point in the same.

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 55 - If a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.

Page 53 - 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 318 - Again ; the mathematical postulate, that " things which are equal to the same are equal to one another," is similar to the form of the syllogism in logic, which unites things agreeing in the middle term.

Page 21 - The angles which one straight line makes with another upon one tide 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.

Page 27 - To construct a triangle of which the sides shall be equal to three given straight lines ; but any two whatever of these lines must be greater than the third (20.