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Joseph-Louis Lagrange (1736–1813) found mathematical beauty in many of the fields in which he worked. Here I present some examples from solid geometry.

He considered beautiful Albert Girard’s (1595–1632) theorem relating the area and the angles of a spherical triangle:

The area of a triangle ABC on the surface of a unit sphere is A + B + C − π.

Jean-Étienne Montucla (1725–99) had earlier called the result ‘very elegant’, but had complained that Girard had proved it in a ‘quite laborious and obscure’ fashion. Lagrange thought John Wallis’s (1616–1703) proof was beautiful.

Another result that Lagrange admired was the following:

In any stereographic projection of a sphere onto a plane, any circle on the sphere that does not pass through the point of projection is projected to a circle on the plane (see attached image).

Hence to find the image of such a circle under projection it suffices to find the images of three distinct points on the circle. This fact Lagrange thought a ‘beautiful property of the stereographic projection’.

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#Lagrange #geometry #SphericalGeometry #SolidGeometry #Montucla #MathematicalBeauty #elegance