# Semicubical parabola

From JSXGraph Wiki

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();
```