Daniel Curtin

Around 1746, Euler took up the problem of the surface area of scalene cones, cones in which the vertex does not lie over the center of the base circle. Calling earlier solutions by Varignon and Leibniz insightful but incomplete and extending his solution to conical bodies with noncircular bases, Euler published his results in 1750. He had not actually calculated any particular areas—not surprisingly, as they generally lead to elliptic integrals. Instead, he showed how to reduce the problem to calculating the arclength of certain curves, carefully elucidating the many ways these curves may be defined. Although the curves seem naturally to involve transcendental quantities, he showed how to adjust so only algebraic quantities are needed. Some details of Euler’s solution for the scalene cones are presented here.