Treatise on Conic Sections |
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Apollonius Archimedes Archytas Aristaeus asymptotes axes axial triangle bisected Book CD² central conic centre circle cone conic sections conjugate diameters conjugate hyperbola construction corresponding CP² determined draw drawn parallel ellipse equal equation Euclid Eutocius figure follows four-line locus geometrical given Hence intersection Join latus rectum M₂H meet the curve Menaechmus middle point normal opposite branch ordinate P₁ Pappus parabola parallel to QQ parallelogram parameter perpendicular plane PN² problem Proclus produced proof Prop Proposition proved quadrilateral QV² ratio rectangle reference respectively right angles right cone segment similar triangles similarly solution square Suppose tangent at Q theorems treatise vertex whence αἱ ἄρα δὲ ἐπὶ ἔχει καὶ πρὸς τὸ ἀπὸ τὰ τῇ τὴν τῆς τοῦ τῷ τῶν ὑπὸ ὡς
Popular passages
Page 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...
Page cii - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.
Page 214 - AB describe a segment of a circle containing an angle equal to the given angle, (in.
Page 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...
Page 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...
Page 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.
Page 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.
Page 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.
Page 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.