 ## Solutio Problematis ad Geometriam Situs Pertinentis.

First edition of “one of Euler's most famous papers—the Königsberg bridge problem. It is often cited as the earliest paper in both topology and graph theory” (Euler Archive, E53). The city of Königsberg is situated on the River Pregel, across which seven bridges had been built in Euler’s time, most of which connected to the island of Kneiphof; the problem asked whether it was possible to devise a route that would allow one to cross each of the bridges exactly once. “When reading Euler’s original proof, one discovers a relatively simple and easily understandable work of mathematics; however, it is not the actual proof but the intermediate steps that make this problem famous. Euler’s great innovation was in viewing the Königsberg bridge problem abstractly, by using lines and letters to represent the larger situation of landmasses and bridges. He used capital letters to represent landmasses, and lowercase letters to represent bridges. This was a completely new type of thinking for the time, and in his paper, Euler accidentally sparked a new branch of mathematics called graph theory, where a graph is simply a collection of vertices and edges. Today a path in a graph, which contains each edge of the graph once and only once, is called an Eulerian path, because of this problem. From the time Euler solved this problem to today, graph theory has become an important branch of mathematics, which guides the basis of our thinking about networks” (MAA). Although Euler felt that the Königsberg bridge problem was trivial, he was still intrigued by it, and believed it was related to Leibniz’s geometria situs, or geometry of position, although today Leibniz’s ideas are viewed as an anticipation of the subject of topology, whereas the bridge problem is one of graph theory. After Euler’s paper, graph theory developed rapidly with major contributions made by Augustin-Louis Cauchy, William Rowan Hamilton, Arthur Cayley, and Gustav Kirchhoff, among many others.

“In March 1736 Karl Ehler, the mayor of Dantzig (now Gdansk), a city eighty miles from Königsberg, imparted to Euler his thoughts on a recreational puzzle about the seven bridge of Konigsberg. It was part of their ongoing correspondence, which covered such items as artillery, real and imaginary numbers, and the rectification of curves. Ehler called the bridges problem ‘an outstanding example of the calculus of position.’ Euler had already solved it. The city of Konigsberg in East Prussia (now Kaliningrad in Russia) comprises four sections. At the center is an island in the Pregel River, and in Euler’s time seven bridges spanning the river connected the island with the other three sections. The question was whether someone could pass over the bridges in a connected walk, crossing each bridge once, and return to the same spot. The puzzle itself, unrelated to Euler’s mathematical research, was among several problems that he addressed only once. While Leibniz and Wolff posed problems of this type, Euler seems to have learned of them from Johann I Bernoulli. Finding the Königsberg Bridge Problem simple, Euler solved it negatively – not with mathematics but with reasoning alone. The article ‘Solutio problematis ad geometriam situs pertinentis’ (Solution of a problem relating to the geometry of position) gave his conclusion. Submitted the next year for volume 8 of the Commentarii, it was not published until 1741 in an issue containing thirteen mathematical articles – two by Daniel Bernoulli and eleven by Euler [see below].

“’Solutio problematis’ contains no graphs but is considered the first work in graph theory. The requisite type of graphs to represent the possible Königsberg walk under the given conditions did not appear until the nineteenth century. Euler divided this paper into twenty-one numbered paragraphs. After paragraph 3 rejects as unworkable any attempt to solve the problem by checking all possible paths, the paper considers the transit entrances to land regions rather than the crossing of bridges” (Calinger, pp. 130-131). The four regions were denoted by the capital letters A, B, C, D (A being the island) and the seven bridges by lower case letters a, b, c, d, e, f, g: a & b connected A to B; c & d connected A to C; e connected A to B; f connected B to D and g connected C to D. “Euler noted that, if a region had an odd number of bridges (k), then the letter of that region must appear (k + 1)/2 times in the string of capital letters that represent the entire journey … Since five bridges connect the island to the city’s other regions, the frequency [of A in the tour] will be (5 + 1)/2 = 3. The frequency for B, C and D, there being three bridges, is (3 + 1)/2 = 2. The sum of these frequencies is nine [3 + 2 + 2 + 2], but the sum for a path crossing each of the seven bridges only once is eight [because each bridge separates two regions]; the Königsberg tour under the given conditions is this impossible; the problem has no solution” (ibid.).

Of the other ten papers by Euler in this volume, the most significant is ‘Theorematum quorundam ad numeros primos spectantium demonstratio’ (pp. 141-146), in which Euler proves ‘Fermat’s Little Theorem.’ This was first stated the theorem in a letter dated October 18, 1640, to his friend and confidant Frénicle de Bessy. Fermat did not send Frénicle the proof, only writing ‘I would send you a demonstration of it, if I did not fear going on for too long.’ Euler was the first to publish a proof, in the present paper.

“Number theory continued to be Euler’s passion and a wellspring of challenging problems; its higher degrees of abstraction attracted him. Among the circle of scholars he met or corresponded regularly with were Goldbach and Krafft, both of whom particularly discussed number theory with him, and by 1736 Euler was inventing ways to prove its theorems by introducing the concepts, definitions, and methods required to complete these theorems; he assiduously tried to consolidate the methods.

“After beginning with studies of 2p - 1 – 1, in 1736 Euler stated )without modern notation) Fermat’s Little Theorem: [if p denotes a prime number and a any whole number not divisible by p], then in modern notation ap - 1 – 1 is divisible by p. In ‘Theorematum quorundam ad numeros primos spectantium demonstratio’ (A proof of certain theorems regarding prime numbers), Euler gives his first of four proofs of the theorem, a clumsy additive one based on mathematical induction on the notation a … His proof employs the binomial expansion of (1 + 1)p - 1 – 1, the subtraction of consecutive binomial coefficients and their divisibility, and an appropriate rearrangement of terms” (Calinger, p. 135).

The remaining nine Euler papers in the present volume are as follows (with comments based on the Euler Archive).

Methodus universalis serierum convergentium summas quam proxime inveniendi (Universal methods of series), pp. 3-9. Euler evaluates the first ten terms of

z(2) = 1/12 + 1/22 + …. + 1/102

to be 1.549768 (probably in his head) and gives an expression for the error term. Then he finds the sum of the first million terms of the harmonic series

1/1 + 1/2 + 1/3 + …..

to be 14.392669.

Inventio summae cuiusque seriei ex dato termino generali (Finding the sum of any series from a given general term). Further studies on z(2), and an infinite series approximation to for the sum of the first n terms of the harmonic series.

Investigatio binarum curvarum, quarum arcus eidem abscissae respondentes summam algebraicam constituant (Investigation of pairs of curves whose arcs that correspond to the same abscissa constitute an algebraic sum).

De oscillationibus fili flexilis quotcunque pondusculis onusti (On the oscillations of a flexible wire weighted with arbitrarily many little weights).

Methodus computandi aequationem meridiei (A method for computing the equation of a meridian).

De constructione aequationum ope motus tractorii aliisque ad methodum tangentium inversam pertinentibus (On the construction of equations using dragged motion, and of other things pertinent to the inverse method of tangents).

Solutio problematum rectivicationem ellipsis requirentium (Solution of a problem requiring the rectification of an ellipse). “Euler starts with integrals of a certain form, which are really elliptical integrals, and derives second-order ordinary differential equations using the so-called “Modular equation” whose solution can be put back through the given integral. Then several geometric problems are solved, which cause special cases of derived differential equations to appear.”

Methodus universalis series summandi ulterius promota (Universal method for summation of series, further developed). Euler considers power series of the form

f(0) + f(1)x + f(2)x2 + f(3)x3 + ...,

Curvarum maximi minive proprietate gaudientium inventio nova et facilis (New and easy method of finding curves enjoying a maximal or minimal property).

The volume also contains two papers by Daniel Bernoulli on hydrodynamics (pp. 99-112 & 113-127).

Calinger, Leonhard Euler, 2016. ‘Leonhard Euler’s solution to the Konigsberg Bridge Problem,’ https://www.maa.org/press/periodicals/convergence/leonard-eulers-solution-to-the-konigsberg-bridge-problem

Pp. 128-140 and Plate VIII in: in Commentarii academiae scientiarum Petropolitanae Tomus VIII ad annum MDCCXXXVI. 4to (259 x 206 mm), pp. [vi],  4-452 with 24 folding engraved plates. Contemporary speckled calf, rebacked.

Item #4791

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