## 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 |

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**wherefore**AL and BC are each of them equal to BG ; and things that are equal to the same are equal to one another ; therefore the straight line AL is equal to BC .**Wherefore**from the given point A a straight line AL has been drawn equal ... Page 15

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**wherefore**also the point C shall coincide with the point F , because the straight line AC is equal to DF : but the point B coincides with the point E ;**wherefore**the base BC shall coincide with the base EF , because the point B ... Page 19

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**wherefore**likewise the angle BAC coincides with the angle EDF , and is equal ( 8. Ax . ) to it . Therefore , if two triangles , & c . Q. E. D. PROP . IX . PROB . To bisect a given rectilineal angle , that is , to divide it into two ... Page 23

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**Wherefore**, if at a point , & c . Q. E. D. PROP . XV . THEOR . IF two straight lines cut one another , the vertical or opposite angles shall be equal . Let the two straight lines AB , CD cut one another in the point E ; the angle AEC ... Page 31

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**wherefore**also the angle BCG is equal to the angle BCA , the less to the greater , which is impossible ; therefore ...**wherefore**BC is not unequal to EF , that is , it is equal to it ; and AB is equal to DE ; therefore the two AB , BC ...### 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

altitude angle ABC angle BAC base BC BC is equal 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 parallelogram AC perpendicular point F polygon prism proportionals proposition Q. E. D. PROP radius ratio of AE rectangle CB rectangle contained rectilineal figure remaining angle right angles segment sides BA similar sine solid angle solid parallelopipeds square of AC straight line AB straight line BC tangent THEOR third triangle ABC triplicate ratio vertex wherefore