Elementa doctrinae solidorum; Demonstratio nonnullarum insignium proprietatum quibus solida hedris planis inclusa suntpraedita, pp. 109-140, 140-160, and Tab. II & III, in: Novi Commentarii Academiae Scientiarum Petropolitanæ 4, 1752/53. EULER'S POLYHEDRON FORMULA

EULER, Leonhard

Verlag: St. Petersburg: Typis Academiae Scientiarum, 1758

First edition of the two landmark papers in which Euler proposed and proved his polyhedron formula, one of the most famous formulas in mathematics. These papers constitute the birth of both graph theory and topology. 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" (Francese & Richeson, "The Flaw in Euler's Proof of his Polyhedral Formula", American Mathematical Monthly 114 (2007), pp. 286-296). The formula states that if a polyhedron - a solid body with planar surfaces, such as a cube or dodecahedron (though not necessarily regular) - has V vertices, E edges, and F faces, then V-E+F=2. Euler stated his theorem in his article 'Elementa doctrinae solidorum,' written in 1750. The article actually is an attempt (of which the formula only constitutes a part) to build up stereometry as a three-dimensional analog of plane geometry in a systematic way. In fact, Euler argues that a systematic treatment of stereometry has been almost entirely neglected by all geometers before him while for plane geometry the same thing has been achieved almost in perfection. The main feature of what he calls a systematic treatment seems to be a classification of polyhedra, of which he presents some first steps. The second article, written in 1751 and entitled 'Demonstratio nonnullarum insignium proprietatum quibus solida hedris planis inclusa sunt praedita,' contains what Euler believed to be a demonstration of his formula. This article is well known and broadly discussed in the literature, mostly for the reason that Euler's proof as it stands later turnedout to contain a gap and that in order to get a true theorem, one has to impose certain restrictions on the concept of polyhedron involved (one requirement is convexity, for instance). However, there are other points of significance in this article that are less discussed. For instance, it is worth noting that Euler's tentative proof is largely based on a spatial analogue of the theorem concerning the sum of angles in a triangle in plane Euclidean geometry; this is a concrete expression of his overall approach exposed in the first article. Another interesting feature is the frankness with which Euler furnishes not only proofs of his results but also the heuristics by which he arrived at them in every detail; compare this to the policy of concealment still practiced some generations earlier, by, e.g., Newton, Leibniz and the Bernoullis. This volume contains three other papers by Euler: 'De numeris, qui sunt aggregata duorum quadratorum' (pp. 3-40); 'De constructione aptissima molarum alatarum. (pp. 41-108); 'De motu corporum coelestium a viribus quibuscunque perturbato' (pp. 161-196). Eneström 230 & 231. Thick 4to, pp. [4], 3-494, with 14 folding plates (first few leaves solied). Contemporary calf, spine gilt with lettering piece, marbled paste-downs (rubbed, joints cracked but holding, about half of spine missing). This is probably a thick paper copy - the text block is about 50% thicker than other copies we have seen. Bestandsnummer des Verkäufers ABE-1603717585605

Bibliografische Details

Titel: Elementa doctrinae solidorum; Demonstratio ...
Verlag: St. Petersburg: Typis Academiae Scientiarum
Erscheinungsdatum: 1758
Einband: Hardcover
Zustand: Good
Auflage: 1st Edition