Descartes vs Fermat: the combat of the 17th century

Last week, we contemplated the three circles with radii 1, 2, and 3, tangent to each other. Their centers would lie at the vertices of a triangle with sides 3, 4, and 5 (1+2, 1+3, and 2+3), which forms a right triangle. This setup allows for a straightforward drawing using only a compass and a non-graduated ruler, though a steady hand can make the ruler optional. We start by drawing two perpendicular lines. From their intersection point, we mark a point on one line at three arbitrary units and on the other line at four units. These points serve as the centers of the three tangent circles: the intersection point is the center of the circle with a radius of 1, and the other two points are the centers of the circles with radii 2 and 3, respectively. Now, it might seem easier to find the radii of the two circles tangent to the other three, one externally and one internally, but is it really?

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Joan Brossa, with ‘Ou amb dos Rovells,’ explored the intersection of poetry and mathematics. The crucial verses of Soddy’s poem, concerning Descartes’ theorem, state, “it is the addition of their squares / half the square of the sum.” This equation expresses that the sum of the squares of the curvatures equals half the square of the sum of these curvatures (recall that curvature of a circle is the reciprocal of its radius). Denoting Q, R, S, and T as the respective curvatures of the four tangent circles:

[Q^2 + R^2 + S^2 + T^2 = \frac{1}{2} (Q + R + S + T)^2]

For the circles with radii 1, 2, and 3, if we denote r as the radius of the fourth circle tangent to these three, we have:

[Q = 1]

[R = \frac{1}{2}]

[S = \frac{1}{3}]

[T = \frac{1}{r}]


[1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{r^2} = \frac{1}{2} \left(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{r}\right)^2]

From which we can deduce the value of r (as it is a quadratic equation, we will obtain two values, one for the inner circle and one for the outer). It’s manageable but tedious, so it’s beneficial to resort to a simple formula that calculates the value of the fourth radius in terms of the other three (can you find it?).

In the second part of Soddy’s erotic-mathematical poem, extending the theorem to the three-dimensional case of five tangent spheres, the key verses are “it is the square of the sum / three times the sum of squares,” which translates to:

[Q^2 + R^2 + S^2 + T^2 + V^2 = \frac{1}{3}(Q + R + S + T + V)^2]

Philosopher vs. Lawyer

We can’t discuss Descartes, especially after talking about analytical geometry, without mentioning Pierre de Fermat, the other great French mathematician of the time (who, in reality, was a lawyer who dabbled in math in his spare time, earning him the title “prince of amateurs”), as he discovered analytical geometry before Descartes did. If today we speak of Cartesian coordinates instead of Fermatian ones, it’s only due to Descartes’ greater prestige and his presentation of the material in a clearer and more systematic way.

Annoyed by Fermat’s preemptive strike, Descartes attempted to discredit the “incompetent lawyer,” criticizing his methods as lacking rigor. But faced with Fermat’s impressive results, Descartes eventually conceded, writing to him: “In view of the latest method you’ve used to find tangents of curved lines, I can only say it’s very good, and if you had explained it this way from the beginning, I wouldn’t have questioned it at all.”

A somewhat deceptive display of fair play, as behind the scenes, both heavyweight mathematicians were critical of each other at every opportunity (as Mersenne, the third great mathematician of the time, knew all too well, being inadvertently drawn into their disputes). It was a multi-round bout where, despite Descartes lending his name to coordinates, neither emerged victorious. This is partly because, as we’ve seen, the trophy for analytical geometry had been claimed long before by Apollonius of Perga and Omar Khayyam.

To continue this discussion in English, let’s delve deeper into the math behind the tangent circles and explore the historical context of Descartes and Fermat’s rivalry.