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 ${\displaystyle x=r\sin \theta ,\ \ y=r\sin \theta .}$ 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

${\displaystyle V=\int _{0}^{\pi }A(\theta )\ d\theta .}$

For a cylindrical ungula (radius 1), the intersection is at z = k sin θ, and ${\displaystyle A={\frac {k}{2}}\sin \theta .}$ Then

${\displaystyle V={\frac {k}{2}}\int _{0}^{\pi }\sin \theta \ d\theta \ =\ -{\frac {k}{2}}\cos \theta \ {\Bigg \vert }_{0}^{\pi }=k.}$.

For a cone of height h on the unit disk, the intersection is found from ${\displaystyle h(1-r)=kr\sin \theta \ \ {\text{giving}}\ \ r=(1+{\frac {k}{h}}\sin \theta )^{-1}.}$ Using the area of a triangle with base 1 and height z produces a volume

${\displaystyle V={\frac {h}{2}}\int _{0}^{\pi }{\frac {\sin \theta \ d\theta }{1+{\frac {k}{h}}\sin \theta }}.}$

References

1. ^ Websters Revised Unabridged Dictionary (1913)
2. ^ Gregory of St. Vincent (1647) Opus Geometricum quadraturae circuli et sectionum coni
3. ^ Blaise Pascal Lettre de Dettonville a Carcavi describes the onglet and double onglet, link from HathiTrust
4. ^ 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