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Apollonius appear applied Archimedes asymptotes axes axis base bisected Book branch centre chord circle complete cone conic conic sections conjugate constant construction contained corresponding curve described determined diameter direction draw drawn ellipse equal equation Euclid expression extremity fact fall figure fixed follows further geometrical given gives greater Hence hyperbola increases intersection Join length less locus manner means meet method moves normal opposite branch ordinate original Pappus parabola parallel parallelogram parameter particular passing perpendicular plane position possible problem produced proof Prop proportion Proposition proved quadrilateral ratio rectangle reference relation respectively result right angles segment separate side similar similar triangles similarly solution square straight line Suppose taken tangent touch treatise triangles whence προς υπό
Pagina lviii - ... spheres are to one another in the triplicate ratio of their diameters, and further that every pyramid is one third part of the prism which has the same base with the pyramid and equal height; also, that every cone is one third part of the cylinder having the same base as the cone and equal height they proved by assuming a certain lemma similar to that aforesaid. And, in the result, each of the aforesaid theorems has been accepted* no less than those proved without the lemma. As therefore my work...
Pagina 214 - AB describe a segment of a circle containing an angle equal to the given angle, (in.
Pagina cviii - To a given straight line to apply a parallelogram equal to a given rectilineal figure, and deficient by a parallelogram similar to a given parallelogram...
Pagina xlv - Take a cone with vertex 0 whose surface passes through the circle or ellipse just drawn. This is possible even when the curve is an ellipse, because the line from 0 to the middle point of AD is perpendicular to the plane of the ellipse, and the construction is effected by means of Prop. 7. Let P be any point on the given ellipse, and we have only to prove that P lies on the surface of the cone so described, Draw PN perpendicular to AA'. Join ON, and produce it to meet AD in M. Through M draw HK parallel...
Pagina lxxi - Euclid did not work out the synthesis of the locus with respect to three and four lines, but only a chance portion of it, and that not successfully ; for it was not possible for the said synthesis to be completed without the aid of the additional theorems discovered by me.
Pagina cv - ... be given, they shall each of them be given. Let AB, BC contain the parallelogram AC given in magnitude, in the given angle ABC, and let the excess of BC above AB be given ; each of the straight lines AB• BC is given.
Pagina xxiv - A cone is a solid figure described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed.
Pagina cvii - XXVIII. and XXIX. B. VI. These two problems, to the first of which the 27th prop. is necessary, are the most general and useful of all in the Elements, and are most frequently made use of by the ancient geometers in the solution of other problems ; and therefore are very ignorantly left out by Tacquet and Dechales in their editions of the Elements, who pretend that they are scarce of any use.