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 |
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Page 25
... angles CBA , ABD : take away the common angle ABC , the remaining angle ABE is equal to the remaining angle b 3. Ax , ABD , the less to the greater , which is impossible ; therefore BE is not in the same straight line with BC . And , in ...
... angles CBA , ABD : take away the common angle ABC , the remaining angle ABE is equal to the remaining angle b 3. Ax , ABD , the less to the greater , which is impossible ; therefore BE is not in the same straight line with BC . And , in ...
Page 26
... ABC be a triangle , and let its side BC be produced to D ; the exterior angle ACD is greater than either of the interior op- posite angles CBA , BAC . Bisect a AC in E , join BE and ... angle ACB ; therefore the angles ACD 26 THE ELEMENTS.
... ABC be a triangle , and let its side BC be produced to D ; the exterior angle ACD is greater than either of the interior op- posite angles CBA , BAC . Bisect a AC in E , join BE and ... angle ACB ; therefore the angles ACD 26 THE ELEMENTS.
Page 27
... ABC , ACB ; but ACD , ACB are to- gether equal to two right angles ; therefore the angles ABC , b 13. 1 . BCA are ... angle . Let ABC be a triangle , of which the side AC is greater than the side AB ; the angle ABC is also greater than ...
... ABC , ACB ; but ACD , ACB are to- gether equal to two right angles ; therefore the angles ABC , b 13. 1 . BCA are ... angle . Let ABC be a triangle , of which the side AC is greater than the side AB ; the angle ABC is also greater than ...
Page 28
... angle , & c . Q. E. D. See N. a 3. 1 . PROP . XX . THEOR . ANY two sides of a triangle are together greater than the third side . Let ABC be a triangle ; any two sides of it together are greater than the third side , viz . the sides BA ...
... angle , & c . Q. E. D. See N. a 3. 1 . PROP . XX . THEOR . ANY two sides of a triangle are together greater than the third side . Let ABC be a triangle ; any two sides of it together are greater than the third side , viz . the sides BA ...
Page 33
... angle GBC is equal to the angle DEF ; therefore the base GC is equal a to the base DF , and the triangle GBC to the ... ABC is equal to the angle DEF ; the base therefore AC is equal a to the base DF , and the third angle BAC to the ...
... angle GBC is equal to the angle DEF ; therefore the base GC is equal a to the base DF , and the triangle GBC to the ... ABC is equal to the angle DEF ; the base therefore AC is equal a to the base DF , and the third angle BAC to the ...
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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.