Now we come to Chapter 20: Leibniz's Quest for Descartes' Secret. Leibniz was attracted to aspects of Descartes' philosophy but was seriously repelled by it as well. Leibniz was critical of Descartes' principle of doubt, suggesting that degrees of doubt rather than absolute doubt be admitted in specific cases (209).
Some of Leibniz's major interests are outlined. I note a mutual interest with Descartes in Ramón Llull's ars combinatoria (210). After three years in Paris, facing the prospect of being recalled to Hanover, Leibniz urgently pursued his aim of inspecting everything that Descartes ever wrote. On June 1, 1676 he succeeded in gaining permission to view Descartes' hidden manuscripts. Scanning the Preambles, Leibniz, a Rosicrucian, recognized an oblique reference to the Rosicrucians (213). The secret notebook, De solidorum elementis, contained obscure formulas and figures. The geometrical figures were depictions of the five Platonic solids, and a connection to mysticism was evident. Leibniz began to copy the records, recognized what was going on, and added a marginal note (219).
Descartes' notebook disappeared, and Leibniz's papers on this subject remained undetected for two centuries. Several subsequent viewers of these documents failed to crack the code. Finally, in 1987, Peter Costabel published his analysis of Leibniz's copy of Descartes' manuscript (220). Leibniz had discovered that Descartes discovered a formula that generalizes the structural characteristics of the Platonic solids (221).
Chapter 21: Leibniz Breaks Descartes' Code and Solves the Mystery. Kepler had postulated a connection between the five Platonic solids and the spacing of the six known planets. Descartes found a formula for all polyhedra, but because others would connect this with Kepler and Copernicus, and so kept it to himself (225-229). Descartes' formula F + V - E = 2 inaugurates the field of topology. Euler discovered this formula, which was named after him.
Other misfortunes befell Descartes' legacy in the 17th century, when his works were proscribed by the Catholic Church and teaching of Cartesian philosophy banned in France. It wasn't until 1824 that his works were reprinted. Adrien Baillet came close to crediting Descartes' discoveries in his biography, but not being a mathematician, did not understand Leibniz's explanation and omitted publishing the information (230). Leibniz remained obsessed and ambivalent concerning Descartes, praising him while alleging limitations. Leibniz kept in contact with Cartesian scholars (231). Leibniz was at work developing the calculus. Concerned about the priority dispute with Newton, Leibniz would not have wanted to acknowledge an influence from Descartes (234-235).
Aczel adds an epilogue to this story. Descartes is seen as the great forerunner of contemporary astrophysics, heavily dependent on geometry linked to algebraic methods. The Platonic solids are n longer relevant, but . . . but satellite data obtained in 2001 supports the notion that the geometry of the universe as a whole fits the geometry of some of the Platonic solids (238-239). One new model posits the universe as an octahedron folded onto itself. The icosahedron and dodecahedron have also served as models.
It's a somewhat peculiar final tribute to Descartes, and Descartes' whole life story is a somewhat roundabout way of getting to discussing the mysterious notebook, but the story is nonetheless interesting, and, aside from the tribute to the mathematical and scientific geniuses of the early modern world, it reveals even more the peculiarities and complexities of the Enlightenment and the scientific revolution.
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