CHAPTER 9
SCOOPED BY DESCARTES?
In 1860, over a century after Euler presented his proof of the polyhedron formula, evidence surfaced that René Descartes, the famous philosopher, scientist, and mathematician, had known of this remarkable relationship in 1630, more than one hundred years before Euler. The evidence was found in a long-lost manuscript. The story is fascinating, as is the debate over whose name should accompany the polyhedron formula.
Descartes was born to a noble, if not wealthy, family in 1596 in La Haye, France, just outside of Tours. His mother died a few days after his birth, and his father, although supportive of his “little philosopher,” was absent for much of René’s childhood.
Young René was a sickly boy who developed into a hypochondriac as an adult. As a boy he attended the Jesuit school at La Flèche, and one of his teachers allowed him to remain in bed for as long as needed each morning, even when the other boys were attending lessons. Descartes used this time to think. He continued this practice throughout his life, nurturing many of his greatest ideas during the peaceful and quiet morning hours spent in bed.
A common theme in Descartes’ life was his quest for solitude. As he put it, “I desire only tranquillity and repose.”2 This need for few distractions is reflected in his many relocations and his lifelong bachelorhood. During his stint in the army he enjoyed the extended periods of peace which afforded him quiet times for deep reflection. Although by no means a recluse, Descartes was always yearning for time alone to work on his scientific and philosophical pursuits. His motto illustrated this desire: bene vixit qui bene latuit (he has lived well who has hidden well).
Figure 9.1. René Descartes.
In 1637 Descartes published a short book with a long title, the influential Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences (Discourse on the Method for Rightly Directing One’s Reason and Searching for Truth in the Sciences3). The publication of Discourse on Method marked the beginning of modern philosophy. In this book, which is now considered a literary classic, Descartes outlined a philosophy based on doubt and rationalism. It contains the most famous sentence in philosophy: cogito ergo sum, I think, therefore I am. His philosophy was one of the foundations of the Scientific Revolution.
Discourse on Method contained three appendices, the most important and influential of which, the hundred-page La géométrie (The Geometry), is frequently cited as the birth of analytic geometry, a subject so ingrained in today’s mathematics that it is difficult to imagine working without it. (We should also give credit to Fermat, a contemporary of Descartes, for his contributions in this area.) Analytic geometry is the melding of geometry and algebra. In analytic geometry we introduce a coordinate system, and using it we locate a point by giving its coordinates (x, y). This system of coordinates is now called Cartesian coordinates, after Descartes. The power of this approach is that geometric figures—circles, lines, curves—can be represented by algebraic equations, enabling us to use the tools of algebra to solve geometric problems. Although the analytic geometry found in The Geometry is not fully formed (for instance, nowhere does Descartes explicitly create coordinate axes), many of the key ideas are present.
In 1649, after three years of repeated invitations, the fifty-three-year-old Descartes agreed to visit the young Swedish queen Christina and give her lessons in philosophy. The Queen insisted that the five-hour, thrice-weekly tutorials began at five o’clock in the morning and that they be held in an unusually cold room (during the coldest Swedish winter in sixty years).
The early morning lessons forced Descartes to abandon his long tradition of remaining in bed through the morning hours. These brutal conditions may have weakened Descartes’ already frail body. On February 1, 1650, only months after he arrived in Sweden, he contracted pneumonia. He refused treatment from Christina’s doctor, preferring to prescribe his own—a mixture of wine and tobacco that induced him to vomit the quickly gathering phlegm. His cure proved ineffective. He died on February 11, 1650.
Descartes’ friend, the French ambassador Hector-Pierre Chanut, took it upon himself to ship Descartes’ personal effects back to Paris, where they would be collected by Chanut’s brother-in-law, Claude Clerselier. But the ship wrecked in the Seine, spilling its contents into the river. Descartes’ possessions, including the trunk containing many pages of notes and manuscripts, floated away. Fortunately, three days later the trunk was recovered. The papers were carefully separated and hung to dry like the day’s laundry.
Now in possession of Descartes’ papers, Clerselier began publishing them. He also made the documents available for scholars to examine. Leibniz was one of the mathematicians interested in Descartes’ waterlogged notes. On one of his trips to Paris, Leibniz made a copy of some of Descartes’ notes about polyhedra, which dated back to roughly 1630. These important notes are now called Progymnasmata de solidorum elementis (Exercises in the Elements of Solids).
Clerselier died in 1684, eight years after Leibniz’s visit, leaving manuscripts still unpublished. One of these was The Elements of Solids. The original was never seen again. Leibniz’s personal copy disappeared and was not recovered for nearly two centuries. If not for providence, we would never know of Descartes’ insightful work on polyhedra.
Foucher de Careil, a nineteenth-century Descartes scholar, was aware from Leibniz’s letters that he had made copies of Descartes’ missing manuscripts. In 1860 he searched for these documents in the well-organized Leibniz collection at the Royal Library of Hanover, but he did not find them. In a stroke of amazing good fortune, however, he found a dusty pile of unknown and uncataloged papers belonging to Leibniz in a neglected cupboard. It was in this collection that de Careil found Leibniz’s copy of The Elements of Solids.
Like those who studied polyhedra before him, Descartes’ approach was metric. Many of his formulas dealt with angle measures. Unlike his predecessors, but like Euler a century later, Descartes also took a combinatorial approach to polyhedra: he counted features on a polyhedron and created algebraic relations among them. While Euler would later tally the number of vertices, edges, and faces, finding the relationship V − E + F = 2, Descartes counted vertices (which, like Euler, he called solid angles), faces, and plane angles.
In his notes Descartes presented many facts about polyhedra. He gave no complete proofs, but it is not difficult to see how each formula follows logically from the ones before. The first major theorem generalized to polyhedra the well-known result for polygons that the sum of the exterior angles is 360°. We will discuss this result, now known as Descartes’ formula, in detail in chapter 20. He also gave what may be the first algebraic proof that there are no more than five Platonic solids.
The work culminated in the following equality relating the number of faces, vertices, and plane angles (F, V, and P, respectively):
P = 2F + 2V − 4.
It is because of Descartes’ discovery of this relation that some scholars say Euler’s formula should bear his name. We simply observe that a polyhedron contains twice as many plane angles as edges (for example, a cube has 24 plane angles and 12 edges). That is, if there are E edges, then there are P = 2E plane angles. Substituting 2E for P yields 2E = 2F + 2V − 4. Dividing by two and rearranging terms, we obtain the familiar polyhedron formula.
So the questions arise: Did Descartes discover Euler’s formula? If so, should it bear his name? The discovery of Descartes’ notes sparked a debate that continues to the present day. Important mathematical personalities have disagreed on this topic. Even today, we find books that state emphatically that Descartes did or did not pre-discover Euler’s formula. Of course, we should keep in mind the words of the eminent philosopher Thomas Kuhn (1922–1996), who wrote, “The fact that [the priority question] is asked . . . is a symptom of something askew in the image of science that gives discovery so fundamental a role.”4
Ernest de Jonquières (1820–1901), one of the first and strongest supporters of Descartes, suggested that the theorem be called the Descartes—Euler formula. In 1890 he wrote, “It cannot be denied then that he knew it, since it is a deduction so direct and so simple, we say so intuitive, from the two theorems that he had just stated.”5 Jonquières’ supporters argue that because Euler’s formula follows so obviously from Descartes’ work, either he knew the relationship, or he was close enough that the theorem should bear his name. They contend that if Descartes rewrote this sketch of a manuscript for publication, then he would have formulated the theorem in the now familiar way. Moreover, even if Descartes did not know the exact relationship, he proved a theorem that is logically equivalent to Euler’s. He and Euler simply chose different quantities to relate. Today it is not uncommon for the polyhedron formula to be called the Descartes–Euler formula.
Surprisingly, much of the debate hinges on the concept of an edge of a polyhedron, which, as we have already discussed, was introduced by Euler. This feature is to us an obvious attribute, but it did not have a name in the time of Descartes. If one did look at the edge of a polyhedra, it was only as a side of one of the polygonal faces; the edges were the geometric objects used to create the angles. In order to give the usual form of Euler’s formula, Descartes would have had to invent the notion of an edge.
Those who contend that Descartes did not anticipate Euler’s formula maintain that the introduction of edges into the formula is crucial. As we saw earlier, Euler recognized that the essential importance of the theorem is that it relates zero-dimensional objects (vertices), 1-dimensional objects (edges), and 2-dimensional objects (faces). In the years that followed, Euler’s formula was generalized to become an important theorem of topology. Topologists did not stop at 2-dimensional faces. As we will see in chapters 22 and 23, Poincaré and others extended Euler’s formula to objects of any dimension.
All agree that Descartes was tantalizingly close, yet he failed to make the last important step. Plane angles are not the proper objects to compare to faces and vertices. To obtain the correct formulation he needed to introduce the notion of an edge. To those who say that Descartes must have known about the relationship with edges, critics point out that even the most accomplished mathematician can fail to see obvious consequences of his or her own work. After carefully examining the manuscript, the mathematician Henri Lebesgue wrote, “Descartes did not enunciate the theorem; he did not see it.”6
There is a widely-held, mistaken belief that the objects in mathematics are named after their discoverers, and that when this does not happen, it is akin to plagiarism or falsification of history. Using this standard, Euler has been wronged repeatedly, for many of his discoveries bear the names of others. (There is a oft-repeated quip that “objects in mathematics are named after the first person after Euler to discover them.”) There are countless examples (even in this book) of mathematical objects not named after the discoverer, but after someone who made influential contributions to the subject—perhaps the first person to truly recognize the importance of the discovery. Kuhn notes that, as in this instance, it is often the case that the priority of discovery is not clear-cut. “We so readily assume that discovering, like seeing or touching, should be unequivocally attributable to an individual and to a moment in time. But the latter attribution is always impossible, and the former often is as well . . . Discovering . . . involves recognizing both that something is and what it is.”7 (Recall Waterhouse’s comment that the regular solids where not special until Theaetetus recognized the common trait that bound them together.)
Whether Descartes did or did not pre-discover Euler’s formula is debatable. But because Descartes’ work was never published and he did not recognize the “useful” form of the formula, it is not unreasonable that we continue to call V − E + F = 2 Euler’s formula.