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In this concluding section of the text, the three major milestones marking the history of the exploration of networks are indicated. These are: Leonhard Euler's work (1735), the introduction of random graphs (1950s), and the launch of the large-scale study of complex networks (the end of the 1990s). A list of a few particularly hot and prospective topics in complex networks is presented.

The basic assumption of ‘rational mechanics’ was that all natural philosophy was mechanics, and that, as mechanics was pursued with greater and greater detail and sophistication, the rest of natural philosophy would fall into place around it. The guiding idea, from Varignon and Hermann at the beginning of the eighteenth century, up to d'Alembert and Euler in mid‐century, was that mechanics could be pursued independently of other natural‐philosophical considerations, that it was the one absolutely secure physical discipline because of its mathematical (and effectively a priori) standing. The chapter explores the rational mechanics of d'Alembert and Euler, and questions whether what was proposed in fact had an a priori standing, and whether it was plausible to assume that recalcitrant phenomena such as the refraction of light, the behaviour of fluids, and gravitation could be accounted for by mechanics.

This chapter introduces a new game of tic-tac-toe that fits squarely within the body of work inspired by mathematician Leonhard Euler's findings on the so-called “Graeco-Latin squares” and the surprisingly interesting problem of arranging thirty-six officers of six different ranks and regiments. In his 1782 paper on the subject, Euler begins with the thirty-six-officer problem and ends with a conjecture about the possible sizes of Graeco-Latin squares. The chapter first explains the rules for a game based on Euler's work, and then analyzes it from a game-theoretic perspective to determine winning and drawing strategies. Along the way, the chapter explains Euler's connection to the story.

Throughout his life, the great mathematician Leonhard Euler spent most of his free time on music, to which he devoted his first book. This chapter discusses how he reformulated the ordering of musical intervals on a new mathematical basis. For this purpose, Euler devised a “degree of agreeableness” that numerically indexed musical intervals and chords, replacing ancient canons of numerical simplicity with a new criterion based on pleasure. Euler applied this criterion (and Aristotle’s teachings about the pleasure of tragedy) to argue that minor intervals and chords evoke sadness through their greater numerical complexity, hence lower degree of agreeableness than the major. This work involved extensive attention to ratios and numerical factorization immediately preceding his subsequent interest in continued fractions and number theory. Having devised a new kind of index, Euler was prepared to put forward indices that would address novel problems like the Königsberg bridge problem and the construction of polyhedra, basic concepts of what we now call topology. Throughout the book where various sound examples are referenced, please see http://mitpress.mit.edu/musicandmodernscience (please note that the sound examples should be viewed in Chrome or Safari Web browsers).

Beside his enormous achievements in mathematics, Leonhard Euler was deeply involved in many areas of physics. His early work on music had a direct bearing on his contemporary study of sound, which in due course contributed to his studies of the mechanics of continuous bodies and vibrating bodies. These important advances in continuum and fluid mechanics also moved Euler to advocate a wave theory for light, as against Newton’s emission (particle) theory. He took the analogy with sound so far as to postulate light “overtones” and “undertones” based on the musical theories of Jean-Philippe Rameau, though lacking any experimental justification. Throughout, Euler used the examples of sound and music as exemplars for a new understanding of light and color. Euler’s later musical writings include his reflections on “ancient” versus “modern” music and its use of chords based on the seventh. He also used music as the centerpiece in his popular account of science. Throughout the book where various sound examples are referenced, please see http://mitpress.mit.edu/musicandmodernscience (please note that the sound examples should be viewed in Chrome or Safari Web browsers).

This chapter considers Eulerian graphs, a class of graphs named for the Swiss mathematician Leonhard Euler. It begins with a discussion of the the Königsberg Bridge Problem and its connection to Euler, who presented the first solution of the problem in a 1735 paper. Euler showed that it was impossible to stroll through the city of Königsberg, the capital of German East Prussia, and cross each bridge exactly once. He also mentioned in his paper a problem whose solution uses the geometry of position to which Gottfried Leibniz had referred. The chapter concludes with another problem, the Chinese Postman Problem, which deals with minimizing the length of a round-trip that a letter carrier might take.

This book concludes with an epilogue, which traces the evolution of graph theory, from the conceptualization of the Königsberg Bridge Problem and its generalization by Leonhard Euler, whose solution led to the subject of Eulerian graphs, to the various efforts to solve the Four Color Problem. It considers elements of graph theory found in games and puzzles of the past, and the famous mathematicians involved including Sir William Rowan Hamilton and William Tutte. It also discusses the remarkable increase since the 1960s in the number of mathematicians worldwide devoted to graph theory, along with research journals, books, and monographs that have graph theory as a subject. Finally, it looks at the growth in applications of graph theory dealing with communication and social networks and the Internet in the digital age and the age of technology.

This prologue begins by considering three examples to demonstrate that in order to solve different problems, fundamentally different kinds of search and different types of proof are required. The first example deals with the nature of computation with a walk through the eighteenth-century town of Königsberg (now Kaliningrad), which had seven bridges connecting the two banks of the river Pregel with two islands. A popular puzzle of the time was whether it is possible to walk through the city by crossing each bridge only once. This puzzle was solved by Leonhard Euler in 1736 in the form of a theorem which states that: A connected graph contains an Eulerian cycle if and only if every vertex has even degree. If exactly two vertices have odd degree, it contains an Eulerian path but not an Eulerian cycle. The second example deals with Hamiltonian paths or cycles, while the third involves factoring integers and chess problems. This book explores how to solve problems as efficiently as possible — and how, and why, some problems are extremely hard.

This chapter explains how to find the area of an angle or polyhedron. It begins with a discussion of how to determine the area of a spherical triangle or polygon. The formula for the area of a spherical triangle is named after Albert Girard, a French mathematician who developed a theorem on the areas of spherical triangles, found in his Invention nouvelle. The chapter goes on to consider Euler's polyhedral formula, named after the eighteenth-century mathematician Leonhard Euler, and the geometry of a regular polyhedron. Finally, it describes an approach to finding the proportion of the volume of the unit sphere that the various regular polyhedra occupy.

This chapter examines the place of the “least action” principle in the causal family and its role in in modern science's transition from teleology to causality. It first provides an overview of the principle of sufficient reason, which illustrates the intricate relations between reasons and causes in the seventeenth theory, along with various conceptions of God proposed by thinkers such as Gottfried Wilhelm Leibniz, Baruch Spinoza, Isaac Newton, and René Descartes. The chapter then considers the work of Leonhard Euler and Pierre-Louis Moreau de Maupertuis on the least action principle. Finally, it analyzes the least action principle's reappearance in the probabilistic context of quantum mechanics, taking into account Richard Feynman's ingenious solution to the long-standing philosophical problem of teleology in physics.

Leonhard Euler (1707–1783) invented graph theory in 1735, by solving a puzzle of interest to the inhabitants of Königsberg. The city comprised three distinct land masses, connected by seven bridges. The residents sought a walk through the city that crossed each bridge exactly once but were consistently unable to find one. Euler showed that such a puzzle would have a solution if and only if every land mass was at the origin of an even number of bridges, with at most two exceptions—which could only be at the start or the end of the journey. Modern treatments of the problem capture Euler's reasoning by employing a diagram in which the land masses are represented by dots (called nodes), while the bridges are represented by line segments connecting the nodes (called edges). Such a diagram is referred to as a graph. This chapter uses algebraic graph theory to solve a number of counting problems.

This chapter considers the concept of coloring the vertices of a graph by focusing on the Four Color Problem. It begins with a discussion of three mathematics problems that involve conjecture, attributed to Pierre Fermat, Leonhard Euler, and Christian Goldbach. It then examines one of the most famous problems in mathematics, the Four Color Problem, which addresses the question of whether it is always possible to color the regions of every map with four colors so that neighboring regions are colored differently. After an overview of the origins of the Four Color Problem, the chapter goes on to analyze the Four Color Conjecture, Alfred Bray Kempe's proof of the Four Color Conjecture, and the Five Color Theorem. Finally, it looks at the Four Color Problem in the twentieth century, along with vertex colorings and their applications.