## The Elements of Euclid: 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. viz. the first six books, together with the eleventh and twelfthMathew Carey, and sold by J. Conrad & Company, S. F. Bradford, Birch & Small, and Samuel Etheridge. Printed by T. & G. Palmer, 116, High-Street., 1806 - Trigonometry - 518 pages |

### From inside the book

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**Parallel**straight lines are such as are in the same plane , and which being produced ever so far both ways , do not meet . Book I. POSTULATES . I. LET it be granted that a straight line may be drawn from any one point to any other point ... Page 34

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**parallel**. Let the straight line EF , which falls upon the two straight lines AB , CD , make the alternate angles AEF , EFD equal to one another ; AB is**parallel**to CD . For , if it be not**parallel**, AB and CD being produced shall meet ... Page 35

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**parallel**to CD . Wherefore , if a straight line , & c .. Q. E. D. PROP . XXIX . THEOR . this pro- IF a straight line fall upon two**parallel**straight See the lines , it makes the alternate angles equal to one ano- notes on ther ; and the ... Page 36

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**parallel**to the same straight line are**parallel**to one another . Let AB , CD be each of them**parallel**to EF ; AB is also pa- rallel to CD . . G A B H -F K C D Let the straight line GHK cut AB , EF , CD ; and because GHK cuts the**parallel**... Page 37

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**parallel**to the given Book I. straight line BC . Which was to be done . PROP . XXXII . THEOR . IF a side of any ...**parallel**to the straight line AB ; and be- cause AB is**parallel**to CE , and AC meets them , the alternate angles BAC ...### Other editions - View all

### Common terms and phrases

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

### Popular passages

Page 30 - Any two sides of a triangle are together greater than the third side.

Page 64 - To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, may be equal to the square of the other part.

Page 30 - 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. Let...

Page 59 - PROP. VIII. THEOR. IF a straight line be divided into any two parts, tour 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 28 - If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Page 165 - 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 19 - THE angles at the base of an isosceles triangle are equal to one another : and, if the equal sides be produced, the angles upon the other side of the base shall be equal.

Page 191 - In right angled triangles, the rectilineal figure described upon the side opposite to the right angle, is equal to the similar, and similarly described figures upon the sides containing the right angle.

Page 39 - 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 sidef. For any rectilineal figure ABCDE can be divided into as many triangles as the figure has sides, by drawing straight lines from a point F within the figure to each of its angles.

Page 180 - Therefore, universally, similar rectilineal figures are to one another in the duplicate ratio of their homologous sides.