Ungula
In solid geometry, an ungula is a section or part of a solid of revolution, cut off by a plane oblique to its base.^{[1]} A common instance is the spherical wedge. The term ungula refers to the hoof of a horse, an anatomical feature that defines a class of mammals called ungulates.
The volume of an ungula of a cylinder was calculated by Grégoire de Saint Vincent.^{[2]} Two cylinders with equal radii and perpendicular axes intersect in four double ungulae.^{[3]} The bicylinder formed by the intersection had been measured by Archimedes in The Method of Mechanical Theorems, but the manuscript was lost until 1906.
A historian of calculus described the role of the ungula in integral calculus:
- Grégoire himself was primarily concerned to illustrate by reference to the ungula that volumetric integration could be reduced, through the ductus in planum, to a consideration of geometric relations between the lies of plane figures. The ungula, however, proved a valuable source of inspiration for those who followed him, and who saw in it a means of representing and transforming integrals in many ingenious ways.^{[4]}^{:146}
Given a solid determined by Q(x,y,z) = constant (often a quadratic form), a method of integrating the volume of the ungula determined by the plane z = 0 and the inclined plane z = ky, uses polar coordinates For a fixed θ, r can be found from Q and intersection with z = ky, and an area A(θ) computed. Then the volume of the ungula is
For a cylindrical ungula (radius 1), the intersection is at z = k sin θ, and Then
- .
For a cone of height h on the unit disk, the intersection is found from Using the area of a triangle with base 1 and height z produces a volume
References
- ^ Websters Revised Unabridged Dictionary (1913)
- ^ Gregory of St. Vincent (1647) Opus Geometricum quadraturae circuli et sectionum coni
- ^ Blaise Pascal Lettre de Dettonville a Carcavi describes the onglet and double onglet, link from HathiTrust
- ^ Margaret E. Baron (1969) The Origins of the Infinitesimal Calculus, Pergamon Press, republished 2014 by Elsevier, Google Books preview
- William Vogdes (1861) An Elementary Treatise on Measuration and Practical Geometry via Google Books