. An encyclopaedia of architecture, historical, theoretical, & practical. New ed., rev., portions rewritten, and with additions by Wyatt Papworth. axis, it is called the directrix of the parabola. By the property shown in this theorem, it ajjpears that if any line QM be drawn parallelto the axis, and if FM be joined, the straight line FM is equal to QM ; for QM is eipialto GP. Coroll. 2. Hence, also, the curve is easily described by points. Take AG ecpial to A F,{fig. 447.), and draw a number of lines M, INI perpendicular to tlie axis AP ; then with thedistances GP, GP, &c. as radii, and from
. An encyclopaedia of architecture, historical, theoretical, & practical. New ed., rev., portions rewritten, and with additions by Wyatt Papworth. axis, it is called the directrix of the parabola. By the property shown in this theorem, it ajjpears that if any line QM be drawn parallelto the axis, and if FM be joined, the straight line FM is equal to QM ; for QM is eipialto GP. Coroll. 2. Hence, also, the curve is easily described by points. Take AG ecpial to A F,{fig. 447.), and draw a number of lines M, INI perpendicular to tlie axis AP ; then with thedistances GP, GP, &c. as radii, and from Fas a centre, describe arcs on each side of A P,cutting the lines MiM, WM, &.c. at MIM,&c. ;then through all the points jM, M, jM, & a curve, which will be a parabola. 1100. Theohem V. // a tangent be drawnfrom the vertex of an ordinate to meet the axisjiroduced, the subtangent PT (_^^. 448.) will lie equal to twice the distance of the ordinate Fid. 4i7. FiK. 418. from tlie vertex. If JIT be a tangent at I\I, the extremity of the ordinate PM ; then the sub-tangent PIis e(jual to twice A P. For draw MK parallel to All,. KM : KI::GK::P;KM : KI::PT : : PM::GK : PT;P : PM::PM : A : PT::PM : GK. Then, by Theor. II., And as MKI, TPIM are similar. Therefore, by equality. And by Cor. Theor. 1., Therefore, by equality, But when the ordinates HI and PM coincide, MT will become a tangent, and GK villbecome ecjual to twice PM. Therefore AP : PT::PM : 2PM, orPT = this property is obtained an easy and accurate method of drawing a tangent ;o anypoint of the curve of a parabola. Thus, let it be re- .^ r ((uired to draw a tangent to any point M in the curve. A Produce PA to T {fg. 449.), and draw MP perpendi- / cular to PT, meeting AP in the point P. Make AT / | ecpial to AP, and join MT, which will be the tangentre(]uired. 1101. Theorem VI. The radius vector is equal tothe distance between the focus and the intersection of atangent at the vertex of an ordinate and the
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