The Elements of Euclid: The Errors by which Theon, Or Others, Have Long 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 twelfth |
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Page 17
... opposite angles . Let ABC be a triangle , and let its side BC be produced to D : the exterior angle ACD shall be greater than either of the interior opposite angles CBA , BAC . Bisect * AC in E , join BE and pro- duce it to F , and make ...
... opposite angles . Let ABC be a triangle , and let its side BC be produced to D : the exterior angle ACD shall be greater than either of the interior opposite angles CBA , BAC . Bisect * AC in E , join BE and pro- duce it to F , and make ...
Page 18
... opposite angle ABC ; to each of these add the angle ACB ; therefore the angles ACD , ACB are greater + than the angles ABC , ACB : but ACD , ACB are together equal to two right angles ; therefore the angles ABC , BCA are less than two ...
... opposite angle ABC ; to each of these add the angle ACB ; therefore the angles ACD , ACB are greater + than the angles ABC , ACB : but ACD , ACB are together equal to two right angles ; therefore the angles ABC , BCA are less than two ...
Page 20
... opposite angle , the exte- rior angle BDC of the triangle CDE is greater than CED : for the same reason , the exterior angle CEB of the triangle ABE is greater than BAC : and it has been demonstrated , that the angle BDC is greater than ...
... opposite angle , the exte- rior angle BDC of the triangle CDE is greater than CED : for the same reason , the exterior angle CEB of the triangle ABE is greater than BAC : and it has been demonstrated , that the angle BDC is greater than ...
Page 23
... opposite to equal angles in each ; then shall the other sides be equal , each to each , and also the third angle of the one to the third angle of the other . A G Let ABC , DEF be two triangles which have the angles ABC , BCA_equal to ...
... opposite to equal angles in each ; then shall the other sides be equal , each to each , and also the third angle of the one to the third angle of the other . A G Let ABC , DEF be two triangles which have the angles ABC , BCA_equal to ...
Page 24
... opposite to equal angles in each triangle be equal to one another , viz . AB to DE : likewise in this case , the other sides shall be equal , AC to DF , and BC to EF ; and also the third angle BAC to the third angle EDF . For , if BC be ...
... opposite to equal angles in each triangle be equal to one another , viz . AB to DE : likewise in this case , the other sides shall be equal , AC to DF , and BC to EF ; and also the third angle BAC to the third angle EDF . For , if BC be ...
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Common terms and phrases
ABC is given AC is equal altitude angle ABC angle BAC base BC bisected centre circle ABCD circumference common logarithm cone Constr 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 logarithm multiple parallel parallelogram AC perpendicular point F polygon prism proportionals proposition pyramid Q. E. D. PROP radius ratio of AE rectangle contained rectilineal figure right angles ROBERT SIMSON segment shewn sides BA similar sine solid angle solid parallelopiped square of AC straight line AB straight line BC tangent THEOR triangle ABC vertex wherefore
Popular passages
Page 141 - 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 40 - If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line, and the several parts of the divided line. Let...
Page 26 - If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite upon the same side, and also the two interior angles upon the same side together equal to two right angles.
Page 46 - If a straight line be divided into any two parts, four 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 - Cor. angles; that is * together with four right angles. There1s, 1. fore all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Page 21 - If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other ; the base of that which has the greater angle shall be greater than the base of the other.
Page 12 - IF two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal; the angle. which is contained by the two sides...
Page 169 - Wherefore, in equal circles &c. QED PROPOSITION B. THEOREM If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.
Page 5 - LET it be granted that a straight line may be drawn from any one point to any other point. 2. That a terminated straight line may be produced to any length in a straight line. 3. And that a circle may be described from any centre, at any distance from that centre.
Page 97 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz.