Elements of Conic Sections |
Common terms and phrases
a²b² a²x² a²y² AC² algebraic expression angle FMI AP X A'P asymptotes axes BC² bisects the angle CA² centre circle circumference CM² CONIC SECTIONS conic surface conjugate axis consequently corresponding abscissas CP² denoted diameter directrix double ordinate drawn perpendicular ellipse equal equation extremities F and F find the algebraic foci focus Geom given line hence hyperbola intersection M'OK meet the asymptotes method of drawing MF'P MP² parabola parallel parallelogram parameter PARM perpendicular to NN point of contact points F polygon Prop Proportion PROPOSITION radii radius vector rectangle right angles semi-conjugate axis similar triangles straight line drawn subnormal subtangent THEOREM transverse axis triangles CAR triangles M'KO triangles MPT verse axis vertex сх
Popular passages
Page 1 - N3 6. A straight line which meets the curve in any point, but which, when produced both ways, does not cut it, is called a tangent to the curve at that point. 7. A straight line drawn from any point in the curve, parallel to the tangent at the vertex of any diameter, and terminated both ways by the curve, is called an ordinate to that diameter.
Page 7 - As the Square of the Transverse Axis : Is to the Square of the Conjugate : : So is the Rectangle of the Abscisses : To the Square of their Ordinate.
Page 31 - The straight line drawn through the for.us, perpendicular to the directrix, is called the axis. The point in which the axis intersects the curve, is called the vertex of the parabola. Thus, BX, drawn through F, perpendicular to EG, is the axis; and A, the middle (Def. I.) of the perpendicular FB, is the vertex of the parabola MAM".
Page 32 - An ordinate to a diameter, is a straight line drawn from any point of the curve to meet that diameter, and is parallel to the tangent at its vertex.
Page 49 - A2 hence the vertices of the transverse axis, hence the square of any ordinate is to the product of its distances from the vertices of the transverse axis as the square of the conjugate axis is to the square of the transverse...
Page 74 - In every ellipsis, the sum of the squares of any two conjugate diameters is equal to the sum of the squares of the two axes.
Page 3 - Take a thread longer than the distance FF', and fasten one of its extremities at F, the other at F'. Then let a pencil be made to glide along the thread so as to keep it always stretched ; the curve described by the point of the pencil will be an ellipse. For, in every position of the pencil, the sum of the distances DF, DF' will be the same, viz., equal to the entire length of the string.
Page 76 - These four points might be projected to the plan on the four edges of the pyramid ; but it is unnecessary to project more than one, since the general principle applies here that if a cone, pyramid, prism or cylinder be cut by a plane parallel to the base, the section is a figure parallel and similar to the base. The one point a...
Page 71 - The square of an ordinate to any diameter, is to the product of the corresponding abscissas, as the square of the conjugate to that diameter, is to the square of the diameter itself.
Page 80 - Scholium 6. It should be remarked, that the common numerator in these values of r, is equal to half the parameter of the transverse axis.