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      on Neile's Parabola

The Semi-cubical Parabola, a.k.a. "Neile's Parabola:

 The First Algebraic Curve to be Rectified


Interesting Facts . . . .

The Semi-cubical Parabola is probably the first curve in our NCB collection whose history is more fascinating than its mathematics.  William Neile (1637-1670) discovered and rectified  - measured - its arc length.  His far more famous professor, John Wallis (1616-1703), published Neile's method in De Cycloide (1659).

Wallis was a charter member of the Royal Society of London that he had helped to organize.  With a Cambridge education followed by his appointment as Savilian Professor of Geometry at Oxford, Wallis was a natural leader in the mathematics community.  In 1655 he published two major papers, one in analytic geometry and the other in infinite analysis.  Most agree these were the two most important research areas of their generation.  When his student, Neile, managed to not only discover a new curve but measure its arc length, Wallis published the results giving Neile full credit.  Neile was only 22.  In 1663, Neile was elected a Fellow of the Royal Society, and thus became both one of its earliest and youngest members.

Later, on the continent, both Leibniz and Huygens investigated the problem of finding a curve down which a particle might descent, under the force of gravity, by falling equal vertical lengths in equal time intervals with an initial velocity different from zero.  In 1673 Huygens' greatest publication, Horologium oscillatorim, presented, among other things, his findings on evolutes and involutes show the evolute of a parabola is a semi-cubical parabola.  The evolute of a curve is the locus of its centers of curvature.

Broad observations are important.  This was the milieu - the mathematical firmament - from which emerged modern physics and calculus.  If this reads like your early physics and calculus education, that is exactly what was in the wake.  To quote E. T. Bell,

"It was inevitable after the work of Cavalieri, Fermat, Wallis, Barrow, and others that the calculus should presently get itself organized as an autonomous discipline.  Like a crystal being dropped into a saturated solution at the critical instant, Newton solidified the suspended ideas of his time, and the calculus took definite shape."
E. T. Bell,  Men of Mathematics

Unfortunately, Neile died in 1670 without knowing that Huygens had bestowed a significant finding upon his work.  He was only 32.

Students at Oxford University are very loyal to their college.  Neile had been at Wadham, the same college as our contemporary today, Sir Roger Penrose.

Equations for the Semi-cubical Parabola - General Equations

have fanciful names resembling botanical terms depending on the relative values of the constant terms
A, B, C, and D.
For the parametric equations

the corresponding Cartesian equation is
and a polar equation is

This particular family of curves is easily entered on a graphing calculator.  May we remind you that to graph in the function mode of
y1 =  ,  you must enter separate formulas for the upper and lower portions of the curve.  In addition to needing an algebraic expression for a function, one must also take a square root with two options ( + ).  Both must be considered.

In this equation the  x-axis is a line of symmetry, but for a good view of the cusp at  (0,0), we suggest you turn off the axes on your graphing calculator.



Arc Length

While calculating the arc length for a semi-cubical parabola gave Neile lasting recognition in the history of mathematics, we no longer use his method to compute.  We apply the techniques of modern calculus which, of course, he could not have known.

Also please note, others had in fact calculated lengths for transcendental (cycloid or logarithmetic spiral) type curves.

See [Gray, pp. 21-22 and Lockwood, p. 11 for a longer discussion.]

Changing the relative values of constant terms often results in graphs fancied to resemble flowers. Thus we find names of "tulip," "hyacinth," "fuchsia," "calyx," and others in the literature.

"The calyx is nothing but the swaddling clothes of the flowers;  the child blossom is bound up in it, hand and foot."

Rushkin,  Ethics of the Dust

Useful Links and Books

John Wallis, F.R.S.

Christian Huygens
Bell, E. T., Men of Mathematics, Simon and Schuster, 1937, p. 118.
Boyer, Carl B., revised by U. C. Merzbach, A History of Mathematics, 2nd ed., John Wiley and Sons, 1991.
Eves, Howard, An Introduction to the History of Mathematics, 6th ed,. The Saunders College Publishing, 1990.
Gray, Alfred,  Modern Differential Geometry of Curves and Surfaces with MATHEMATICA®, 2nd ed., CRC Press, 1998, pp. 21-22.
Katz, Victor J., A History of Mathematics,  PEARSON - Addison Wesley, 2004.
Lockwood, E. H., A Book of Curves, Cambridge University Press, 1961.
Shikin, Eugene V., Handbook and Atlas of Curves, CRC Press, 1995.
Yates, Robert,  CURVES AND THEIR PROPERTIES, The National Council of Teachers of Mathematics, 1952