$$\DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{#1}}$$

Omar Hayyam Method

This plot explains Omar Khayyam's graphical method to solve the cubic equation x3+a2x=b, for a,b>0:
(http://riotorto.users.sourceforge.net/Maxima/gnuplot/omar/index.html)

(Realised with wxmaxima rel. 17.10.1)

 (%i1) load(draw)$$\mbox{}\\\mbox{0 errors, 0 warnings}$  (%i10) a: 3$ b: 26$r: b/(2·a^2)$
 (%i11) draw2d(/· General settings ·/      xrange        = [−3,3],      yrange        = [−2,4],      axis_top      = false,      axis_right    = false,      axis_left     = false,      axis_bottom   = false,      transparent   = true,      user_preamble = "unset tics",      title         = "Omar Khayyam solution to the cubic x^3 + a^2 x = b",      dimensions    = [450,450],      terminal      = svg,      file_name     = "omar",       /· draw parabola y=x^2/a ·/      explicit(x^2/a,x,−3,3),      head_length   = 0.1,      vector([−1.626,2],[−0.5,−0.5]),      label(["y=x^2/a",−1.5,2.1]),       /· draw circle: radius r & center (r,0) ·/      ellipse(r,0,r,r,0,360),      vector([r,0],[0,r]),      label(["r=b/(2 a^2)",r−0.5,r·2/3]),       /· draw axis ·/      points_joined = true,      color         = black,      point_size    = 0,      points([[0,3.5],[0,−1.5]]),      points([[−2.5,0],[2.5,0]]),       /· Mark points P e Q·/      color         = black,      point_type    = 7,      point_size    = 1,      points([[2,4/3]]),      label(["P",2.25,1.42]),      points([[2,0]]),      label(["Q",2.25,0.23]),       /· Draw parallel to parabola axis ·/      point_size    = 0,      color         = blue,      line_type     = dots,      points([[2,2],[2,−0.5]]),       /· Get the solution ·/      line_width    = 4,      color         = red,      line_type     = solid,      points([[0,0],[2,0]]),      label(["The length of this red segment is the solution",1,−0.5])) \$
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Figure 1:E:\Dynamic_Content\Maxima Programs\omar.png

Created with wxMaxima.