Semicubical parabola

A semicubical parabola is a curve defined parametrically as

[math]\displaystyle{ x = t^2 }[/math]

[math]\displaystyle{ y = at^3 }[/math]

From Wikipedia:

The semicubical parabola was discovered in 1657 by William Neile who computed its arc length; it was the first algebraic curve (excluding the line) to be rectified. It is unique in that a particle following its path while being pulled down by gravity travels equal vertical intervals in equal time periods.

From MathDL:

1659:Hendrik van Heuraet sent van Schooten his rectification of the semi-cubical parabola. This was published---his only publication---in the second Latin edition of Descartes' Geometrie. This broke the spell of Aristotle's dictum that curved lines could not in principle be compared with straight lines.

References

• Wikipedia on semicubical parabola
• The MacTutor History of Mathematics archive
• MathDL, the MAA Mathematical Sciences Digital Library

The underlying JavaScript code

var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-1,2,3,-2], keepaspectratio:true, axis:true});
brd.suspendUpdate();
var a = brd.create('slider',[[-0.5,1.8],[1,1.8],[-5,0.20,5]], {name:'a'});

var p = brd.create('curve',
             [function(t){ return t*t;},
              function(t){ return a.Value()*t*t*t;},
              -2, 2
             ],
             {strokeWidth:1, strokeColor:'black'});
brd.unsuspendUpdate();