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 38
... double * of the triangle ABD , because they are upon the same base BD , and between the same parallels BD , AL ; and the square GB is double of the triangle FBC , because these also are upon the same base FB , and * between the same ...
... double * of the triangle ABD , because they are upon the same base BD , and between the same parallels BD , AL ; and the square GB is double of the triangle FBC , because these also are upon the same base FB , and * between the same ...
Page 39
... doubles of equals are equal to one another ; therefore the * 6 Ax . parallelogram BL is equal to the square GB . In the same manner , by joining AE , BK , it can be demon- strated , that the parallelogram CL is equal to the square HC ...
... doubles of equals are equal to one another ; therefore the * 6 Ax . parallelogram BL is equal to the square GB . In the same manner , by joining AE , BK , it can be demon- strated , that the parallelogram CL is equal to the square HC ...
Page 45
... double of AK : but twice the rectangle AB , BC is double of AK , for BK is equal to BC ; therefore the gnomon AKF , together with the square CK , is equal + to twice the rectangle AB , BC : to each of these equals add HF , which is ...
... double of AK : but twice the rectangle AB , BC is double of AK , for BK is equal to BC ; therefore the gnomon AKF , together with the square CK , is equal + to twice the rectangle AB , BC : to each of these equals add HF , which is ...
Page 47
... double of the square of half the line , and of the square of the line between the points of section . Let the straight line AB be divided into two equal parts at the point C , and into two unequal parts at the point D : the squares of ...
... double of the square of half the line , and of the square of the line between the points of section . Let the straight line AB be divided into two equal parts at the point C , and into two unequal parts at the point D : the squares of ...
Page 48
... double of the square of AC : but the square of AE is equal to the squares of AC , CE because ACE is a right angle ; therefore the square of AE is double of the square of AC : again , because EG is equal to GF , the square of EG is equal ...
... double of the square of AC : but the square of AE is equal to the squares of AC , CE because ACE is a right angle ; therefore the square of AE is double of the square of AC : again , because EG is equal to GF , the square of EG is equal ...
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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.