## Elementa Doctrinae Solidorum. WITH: Demonstratio Nonnularum Insignium Proprietatum, quibus Solida Hedris Planis Inclusa Sunt Praedita

*”For his profound and extensive contributions across pure and applied mathematics, Leonhard Euler ranks among the four greatest mathematicians of all time, the other three being Archimedes, Isaac Newton, and Carl Friedrich Gauss." -Ronald S. Calinger, “Leonhard Euler: Mathematica Genius in the Enlightenment” (Princeton 2016)*

FIRST EDITIONS OF TWO LANDMARK PAPERS IN WHICH EULER PROPOSED HIS FAMOUS “POLYHEDRON FORMULA,” ONE OF THE MOST BEAUTIFUL AND CONSEQUENTIAL OF MATHEMATICAL THEOREMS, AND ONE WITH IMPORTANT APPLICATIONS TO TOPOLOGY AND GRAPH THEORY.

Leonhard Euler “not only made decisive and formative contributions to the subjects of geometry, calculus, mechanics, and number theory but also developed methods for solving problems in observational astronomy and demonstrated useful applications of mathematics in technology and public affairs.” (*Brittanica*). One of those contributions, set forth in the two papers offered here, relates to the theory of polyhedra — in essence, solid bodies with planar surfaces, such as a cube or dodecahedron. Indeed, Euler’s result was “the most significant contribution to the theory [of polyhedra] since the foundational work of the ancient Greeks, perhaps the most important contribution ever.” (Christopher Francese and David Richeson, “The Flaw in Euler’s Proof of his Polyhedral Formula”, *American Mathematical Monthly* 114(4)286-296 (2007)). It asserts that the number of vertices, edges, and faces of a polyhedron are related by the formula V – E + F = 2. For example, a cube has eight vertices, twelve edges, and six faces; and 8 – 12 + 6 = 2. In a survey conducted in the fall of 1988 by a mathematical journal, in which readers were asked to rate 24 mathematical theorems on the criterion of “beauty,” Euler’s polyhedron formula rated second, bested only by another Euler discovery, e^{iπ} = – 1. (David Wells, “Are These the Most Beautiful?”, *The Mathematical Intelligencer* 12(3): 37-41 (1990)).

The polyhedron formula might seem at first glance to be a trivial factoid, an insignificant bit of recreational mathematics. In fact it is a profound theorem with surprising, wide-ranging, and important applications. (*See* David S. Richeson, *Euler’s Gem: The Polyhedron Formula and the Birth of Topology* (Princeton 2008)). For example, the theorem can be generalized to yield a method for classifying surfaces (or, more generally, the abstract mathematical objects known as Riemannian manifolds) in terms of a metric known as the “Euler characteristic.” It therefore underlies the Gauss-Bonnet theorem, which relates the overall curvature of a manifold and of its boundary to its Euler characteristic. Euler’s theorem also has a corollary in the mathematical discipline known as graph theory, which deals with the characteristics of sets of nodes connected by “edges,” and which is fundamental to the modern study of social and other networks. The graph-theoretic version of Euler’s formula has important practical applications: “[M]otivated by many problems involving the design of computer chips (integrated circuits), there has been an explosion of research about crossing number problems for graphs in the plane. This involves finding the minimum number of crossings when an abstractly defined graph is drawn in the plane. … Many of these questions involve the use of Euler’s formula to get estimates for the smallest numbers of crossings.” (Joseph Malkevitch, “Euler’s Polyhedral Formula”, available on the American Mathematical Society website.)

Euler’s theorem is one of those mathematical facts that was long hidden in plain sight. “It is remarkable that no one before Euler noticed the polyhedral formula. For centuries the Greeks studied the properties of polyhedra, but they did not notice the relation. With renewed interest in the subject, Kepler and other Renaissance mathematicians studied polyhedra, yet they did not discover the formula. Descartes came very close to discovering the relation, but … he missed a key ingredient. In 1750, Euler wrote to Christian Goldbach, ‘It astonishes me that these general properties of stereometry have not, as far as I know, been noticed by anyone else.” (Francese & Richeson, *op. cit.*). “These mathematicians, and so many others, missed a relationship that is so simple that it can be explained to any schoolchild, yet is so fundamental that it is part of the fabric of modern mathematics.” (Richeson, *op cit.*).

Euler was able to make this breakthrough by abandoning the traditional focus on the *metric* properties of geometric objects (such as distance, area, and volume) and focusing instead on the more abstract properties (such as connectedness or the number of “holes” or “handles” that an object has) that are not affected by distortions in shape or size. (Francese & Richeson, *op. cit.*). (This was similar to the approach that some years earlier had led Euler to his famous solution to the “seven bridges of Königsberg” problem.) Euler’s work on the polyhedron formula and the bridge problem thus laid the conceptual groundwork for the discipline of algebraic topology, a field that is sometimes described as “rubber sheet geometry” since its focus is precisely on properties that remain invariant when an object is stretched, shrunk, or otherwise continuously deformed. Thus, “Euler’s formula marks the beginning of the transition period from geometry to topology.” (Francese & Richeson).

In the first of the two papers, Euler stated his formula, and showed that it held for a number of different types of polyhedra, but did not prove it. The second paper offered a proof — albeit one that turned out to have some flaws. The theorem is nevertheless correct, and the flaws in the proof were subsequently repaired by a later generation of mathematicians.

Offered here is the complete fourth volume of *Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae*, published in 1758 but covering the years 1752 and 1753, containing the two papers described above as well as three other papers by Euler (on representing numbers by the sum of two squares; on the construction of gears; and on the movement of celestial bodies). The *Novi Commmentarii* was published in St. Petersburg from 1750 to 1776 as the journal of Russia’s Imperial Academy of Sciences. The Academy had been founded in 1724 by Tsar Peter I, on the advice of Gottfried Leibniz. It had its ups and downs in the decades following Peter’s death, depending upon the strength and interests of the man or woman on the throne, and on cultural shifts that led Russia to alternately embrace and reject Western European culture and institutions. Not only Euler but other great European scientists and mathematicians, including Daniel and Nicholas Bernoulli, Christian Goldbach, and Nevil Maskelyne, worked at the Academy during its heyday.

The volume includes an engraved title page depicting a heap of fruit above the Society’s motto (“*En addit fructus aetate recentes*”), and fourteen engraved fold-out plates of diagrams and drawings, two of them (II and III) illustrating Euler’s polyhedron papers.

*Provenance:* A printed ownership label bearing the name “Samuel Parr,” cut irregularly from a larger piece of paper, is attached to the front paste-down. This may well be the polymathic minister, schoolmaster, and Whig apologist Samuel Parr (1747-1825), whose wide range of learning is attested to by the contents of his library, as catalogued in *Bibliotheca Parriana: A Catalogue of the Library of the Late Reverend and Learned Samuel Parr, LL.D.* (London 1827). Among the items listed in the *Bibliotheca Parriana* are numerous works in the sciences (including some on mathematics), as well as 23 volumes in total of the *Commentarii Academiae Scientiarum Imperialis Petropolitanae* (as it was known prior to 1747) and the *Novi Commentarii … *(as it was known thereafter), together covering the period from 1746-1763, and thus including the volume offered here.

Elementa Doctrinae Solidorum [Elements of the doctrine of solids], pp. 109-141; Demonstratio Nonnularum Insignium Proprietatum, quibus Solida Hedris Planis Inclusa Sunt Praedita [Proof of some of the properties of solid bodies enclosed by planes], pp. 141-160. Also with the following by Euler: De numeris, qui sunt aggregata duorum quadratorum [On numbers which are the sum of two squares], pp.3-40; De constructione aptissima molarum alatarum [On the construction of the best gears], pp.41-108; De motu corporum coelestium a viribus quibuscunque perturbato [On the movement of celestial bodies perturbed by any number of forces], pp.161-196.

St. Petersburg: Typis Academiae, 1758. Quarto, contemporary calf rebacked, marbled endpapers, edges dyed red. With discreet deaccession stamp from the University of London on the verso of the title page and a penciled notation, perhaps a catalog number, on the front endpaper, but no other library markings. Some browning to introductory material (not affecting Euler) and to edges of last plate. Occasional very faint dampstaining at top margin (not affecting text).

FIRST EDITION OF A LANDMARK MATHEMATICAL THEOREM, WITH AN INTERESTING PROVENANCE.

Price: $2,000 .