Show threadAlan J. CainFeb 26

Spencer expressed enthusiasm for mathematics, and was proud enough of a rather slender theorem he proved in 1840 to reprint the paper as an appendix to his autobiography. (The result had actually been known since the mid-18th century.)

But his geometrical knowledge was actually quite limited, and the statement he gave of Monge’s theorem was carelessly imprecise.

Further, Gaspard Monge’s (1746–1818) original proof is simple and explains the theorem:

Imagine the circles are great circles of three spheres. Then there are two planes that are tangent with all three spheres. (See attached image.) These planes are also in contact with the cones defined by each pair of spheres, and the tangent lines are where these cones pass through the original plane. Thus both these planes contain the apices of the three cones, which are the intersections of pairs of the tangents. That is, the intersection points lie in the intersection of the two planes, which is a straight line.

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#SolidGeometry #MathematicalExplanation #simplicity

Alan J. CainFeb 23

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

NicJan 12

The only thing Flat Earthers have to fear…
… is sphere itself.

#jokeoftheday #dadjokes #SolidGeometry