Greek tragedies had a knack for producing, at just the right moment, a god who had descended from who knows where in an improbable machine to help the protagonists make a timely escape from a delicate situation. What mathematician has not dreamed of inspiration from another world, an unexpected deus ex machina arriving just in time to whisper the brilliant idea that would supply the missing link in his proof?
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Famous and spectacular examples
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There is no need to teach young Carl Friedrich Gauss (1777–1855) a "magic" way to calculate the sum of the first hundred integers: it is said to be the method he used to do so while still at primary school. We owe him this historic deus ex machina. To calculate 1 + 2 + … + 99 + 100, simply add 1 + 100 = 101, then 2 + 99 = 101… all the way to 100 + 1 = 101, giving 1 + 2 + … + 99 + 100 = 50 × 101 = 5,050. The method obviously generalizes to the sum of the first n integers: pair the numbers equidistant from the two ends to obtain 1 + 2 + … + (n – 1) + n = n (n + 1) / 2. The same idea immediately shows that the sum 1 + 3 + … + (2n – 1) of the first n odd integers is simply n2, since here each pair of terms equidistant from the ends sums to 2n. Spectacular, isn't it?
Another result, this time geometric and with an equally spectacular proof, is due to the French mathematician Gaspard Monge (1746–1818), one of the founders of the École polytechnique. Given three circles of different radii whose common external tangents meet pairwise at three points, the aim is to prove that those three points are collinear. Here Monge invoked a deus ex machina that took a detour… through space. Instead of three circles in the plane, he said, imagine three spheres in space resting on a plane P. Rather than tangent lines, consider cones tangent to the spheres in pairs, like ice-cream cones wrapped around double scoops. The vertices of these cones therefore lie in P. There is also another plane that "covers" all three spheres and is tangent to them: the plane Π, obtained by reflecting P across the plane through the centers of the three spheres. Π also contains the vertices of the three cones, which must therefore lie on the intersection of the two planes P and Π. Our result is thus proved by projection!
Today, this property is proved using dilations mapping one circle to another, whose centers are precisely the intersections of the tangent lines. Composing these dilations directly establishes the collinearity of the three centers. Even so, Monge's highly imaginative excursion into space proved fruitful in its day.
