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 twelfthJ. Collingwood, 1827 - 513 σελίδες |
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Σελίδα 5
... , the remainders are equal . IV . If equals be added to unequals , the wholes are unequal . V. If equals be taken from unequals , the remainders are unequal . VI . Things which are double of the same , BOOK I. - DEFINITIONS , ETC. 5.
... , the remainders are equal . IV . If equals be added to unequals , the wholes are unequal . V. If equals be taken from unequals , the remainders are unequal . VI . Things which are double of the same , BOOK I. - DEFINITIONS , ETC. 5.
Σελίδα 6
... double of the same , are equal to one another . VII . Things which are halves of the same , are equal to one another . VIII . Magnitudes which coincide with one another , that is , which exactly fill the same space , are equal to one ...
... double of the same , are equal to one another . VII . Things which are halves of the same , are equal to one another . VIII . Magnitudes which coincide with one another , that is , which exactly fill the same space , are equal to one ...
Σελίδα 30
... double * of the triangle BDC ; and they are therefore equal to one another . But , if the sides AD , EF , opposite to the base BC of the parallelograms ABCD , EBCF , be not termi- * 1 Ax . 2 or 3 Ax . nated 30 EUCLID'S ELEMENTS .
... double * of the triangle BDC ; and they are therefore equal to one another . But , if the sides AD , EF , opposite to the base BC of the parallelograms ABCD , EBCF , be not termi- * 1 Ax . 2 or 3 Ax . nated 30 EUCLID'S ELEMENTS .
Σελίδα 33
... . Q. E. D. PROP . XLI . THEOR . If a parallelogram and a triangle be upon the same base , and between the same parallels ; the parallelogram shall be double of the triangle . * 37. 1 . * 34. 1 . * 10. BOOK I. PROP . XL . XLI . 83.
... . Q. E. D. PROP . XLI . THEOR . If a parallelogram and a triangle be upon the same base , and between the same parallels ; the parallelogram shall be double of the triangle . * 37. 1 . * 34. 1 . * 10. BOOK I. PROP . XL . XLI . 83.
Σελίδα 34
... double of the triangle EBC . * B DE Join AC : then the triangle ABC is equal to the triangle EBC , because they the same base BC , and between are upon the same parallels BC , AE : but the parallelogram ABCD is double of the triangle ...
... double of the triangle EBC . * B DE Join AC : then the triangle ABC is equal to the triangle EBC , because they the same base BC , and between are upon the same parallels BC , AE : but the parallelogram ABCD is double of the triangle ...
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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
Δημοφιλή αποσπάσματα
Σελίδα 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.
Σελίδα 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...
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.
Σελίδα 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...
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.