The original Latin manuscript of
De solidorum elementis was written circa 1630 by Descartes; reviewer
Marjorie Senechal calls it "the first general treatment of polyhedra", Descartes' only work in this area, and unfinished, with its statements disordered and some incorrect. It turned up in
Stockholm in Descartes' estate after his death in 1650, was soaked for three days in the
Seine when the ship carrying it back to Paris was wrecked, and survived long enough for
Gottfried Wilhelm Leibniz to copy it in 1676 before disappearing for good. Leibniz's copy, also lost, was rediscovered in
Hannover around 1860. The first part of
Descartes on Polyhedra relates this history, sketches the biography of Descartes, provides an eleven-page facsimile reproduction of Leibniz's copy, and gives a transcription, English translation, and commentary on this text, including explanations of some of its notation. In
De solidorum elementis, Descartes states (without proof)
Descartes' theorem on total angular defect, a discrete version of the
Gauss–Bonnet theorem according to which the angular defects of the vertices of a
convex polyhedron (the amount by which the angles at that vertex fall short of the 2\pi angle surrounding any point on a flat plane) always sum to exactly 4\pi. Descartes used this theorem to prove that the five
Platonic solids are the only possible regular polyhedra. It is also possible to derive
Euler's formula V-E+F=2 relating the numbers of vertices, edges, and faces of a convex polyhedron from Descartes' theorem, and
De solidorum elementis also includes a formula more closely resembling Euler's relating the number of vertices, faces, and plane angles of a polyhedron. Since the rediscovery of Descartes' manuscript, many scholars have argued that the credit for Euler's formula should go to Descartes rather than to
Leonhard Euler, who published the formula (with an incorrect proof) in 1752. The second part of
Descartes on Polyhedra reviews this debate, and compares the reasoning of Descartes and Euler on these topics. Ultimately, the book concludes that Descartes probably did not discover Euler's formula, and reviewers Senechal and
H. S. M. Coxeter agree, writing that Descartes did not have a concept for the edges of a polyhedron, and without that could not have formulated Euler's formula itself. Subsequently, to this work, it was discovered that
Francesco Maurolico had provided a more direct and much earlier predecessor to the work of Euler, an observation in 1537 (without proof of its more general applicability) that Euler's formula itself holds true for the five Platonic solids. The second part of Descartes' book, and the third part of
Descartes on Polyhedra, connects the theory of polyhedra to
number theory. It concerns
figurate numbers defined by Descartes from polyhedra, generalizing the classical Greek definitions of figurate numbers such as the
square numbers and
triangular numbers from two-dimensional
polygons. In this part Descartes uses both the Platonic solids and some of the
semiregular polyhedra, but not the
snub polyhedra. ==Audience and reception==