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 viii
... whole has been carefully revised throughout . Short Notices of the Lives of EUCLID , and of his ablest translator and commentator , DR . SIMSON , are now , for the first time , prefixed , and are chiefly drawn from Dr. Hutton's ...
... whole has been carefully revised throughout . Short Notices of the Lives of EUCLID , and of his ablest translator and commentator , DR . SIMSON , are now , for the first time , prefixed , and are chiefly drawn from Dr. Hutton's ...
Page 6
... whole is greater than its part . X. Two straight lines cannot inclose a space . XI . All right angles are equal to one another . XII . " If a straight line meet two straight lines , so as to " make the two interior angles on the same ...
... whole is greater than its part . X. Two straight lines cannot inclose a space . XI . All right angles are equal to one another . XII . " If a straight line meet two straight lines , so as to " make the two interior angles on the same ...
Page 9
... whole triangle ABC coincides with the whole triangle DEF , and is equal to it ; and the other angles of the one coincide with the remaining angles of the other , and are equal to them , viz . the angle ABC to the angle DEF , and the ...
... whole triangle ABC coincides with the whole triangle DEF , and is equal to it ; and the other angles of the one coincide with the remaining angles of the other , and are equal to them , viz . the angle ABC to the angle DEF , and the ...
Page 10
... whole AF is equal to the whole AG , of which the parts AB , AC are equal ; the remainder BF is equal to the remainder CG : and FC was proved to be equal to GB ; therefore the two sides BF , FC are equal to the two CG , GB , each to each ...
... whole AF is equal to the whole AG , of which the parts AB , AC are equal ; the remainder BF is equal to the remainder CG : and FC was proved to be equal to GB ; therefore the two sides BF , FC are equal to the two CG , GB , each to each ...
Page 28
... whole exterior angle ACD is equal to the two interior and opposite angles CAB , ABC to each of these equals add the angle ACB , and the angles ACD , ACB are equal to the three angles CBA , BAC , ACB ; but the angles ACD , ACB are equal ...
... whole exterior angle ACD is equal to the two interior and opposite angles CAB , ABC to each of these equals add the angle ACB , and the angles ACD , ACB are equal to the three angles CBA , BAC , ACB ; but the angles ACD , ACB are 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.