## 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 - 518 pages |

### From inside the book

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**multiple**of a less , when the greater is measured by the less , that is , when the greater contains the less a ...**multiple**of the third is also less than that of the fourth ; or , if the**multiple**of the first be equal to that of ... Page 120

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**multiple**of the first is greater than that of the second , but the**multiple**of the third is not greater than the**multiple**of the fourth ; then the first is said to have to the second a greater ratio than the third magni- tude has to the ... Page 123

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**multiple**of a greater magnitude is greater than the same**multiple**of a less . IV . That magnitude of which a**multiple**is greater than the same**multiple**of another , is greater than that other magnitude . PROP . I. THEOR . IF any number ... Page 124

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**multiple**of the second that the third is of the fourth , and the fifth the same**multiple**of the second that the sixth is of the fourth ; then shall the first together with the fifth be the same**multiple**of the second , that the third ... Page 125

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**multiple**of B the second , that C the third is of D the fourth ; and of A , C let the equimultiples EF , GH be taken ; then EF is the same**multiple**of B , that GH is of D. Because EF is the same**multiple**of A , that GH is of C , there ...### 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 parallelogram AC perpendicular point F polygon prism proposition pyramid Q. E. D. PROP radius rectangle CB rectangle contained rectilineal figure remaining angle right angles segment sides BA similar 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 28 - Any two sides of a triangle are together greater than the third side.

Page 62 - 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 28 - 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 57 - 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 26 - If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Page 163 - 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 17 - 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 189 - 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 37 - 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 178 - Therefore, universally, similar rectilineal figures are to one another in the duplicate ratio of their homologous sides.