SEMICUBICAL Parabola
, a curve of the 2d order, of such a nature that the cubes of the ordinates are proportional to the squares of the abscisses. Its equation is ay2 = x3. This curve, AMm, is one of Newton's five diverging parabolas, being his 70th species; having a cusp at its vertex at A. It is otherwise named the Neilian parabola, from the name of the author who first treated of it.
The area of the space APM, is , or 4/15 of the circumscribing rectangle.
The content of the solid generated by the revolution of the space APM about the axis AP, is , or 1/4 of the circumscribing cylinder. And a circle equal to the surface of that solid may be found from the quadrature of an hyperbolic space.
Also the length of any arc AM of the curve may be easily obtained from the quadrature of a space contained under part of the curve of the common parabola, two semiordinates to the axis, and the part of the axis contained between them.
This curve may be described by a continued motion, viz, by fastening the angle of a square in the vertex of a common parabola; and then carrying the intersection of one side of this square and a long ruler (which ruler always moves perpendicularly to the axis of the parabola) along the curve of that parabola. For the intersection of the ruler, and the other side of the square will describe a Semicubical parabola. Maclaurin performs this without a common parabola, in his Geometria Organica.
