## 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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**base BC**shall coincide with the base EF , because the point B coinciding with E , and C with F , if the**base BC**does not coincide with the base EF , two straight lines would inclose a space , which is impossible . Therefore a 10. Ax . the ... Page 18

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**base BC**is common to the two triangles BFC , CGB ; wherefore the triangles are equalb , and their remaining angles , each to each , to which the equal sides are opposite ; therefore the angle FBC is equal to the angle GCB , and the ... Page 19

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**BC**common to both , the two sides DB ,**BC**are equal to to the two AC , CB , each to each ; and the angle DBC is equal to the angle ACB ; therefore the**base**DC is equal to the**base**AB , and the triangle DBC is equal to the triangle ACB ... Page 20

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**base**CD are equal a to one another , but the angle ECD is greater than the angle BCD ; wherefore the angle FDC is likewise greater than BCD ; much more then is the ...**BC**is equal to EF ; therefore**BC**coinciding with 20 THE ELEMENTS. Page 21

... BC is equal to EF ; therefore BC coinciding with EF , BA and Book I. AC shall coincide with ED and DF ; for , if the

... BC is equal to EF ; therefore BC coinciding with EF , BA and Book I. AC shall coincide with ED and DF ; for , if the

**base BC**coin- cides with the base EF , but the sides BA , CA do not coincide with the sides ED , FD , but have a ...### 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.