St. Petersburg: Imperial Academy of Sciences, 1768-1772.
First edition, in the original interim wrappers, of these letters to the Princess Friederike Charlotte Leopoldine Louise of Brandenburg-Schwedt, to whom Euler had given lessons during his stay in Berlin. The work “had an immense success and profoundly influenced contemporary philosophy” (PMM 196, note). “Another multi-volume masterwork by Euler, the Lettres are a principal document of the Enlightenment. Their passion for learning reflects that period's faith in education, including support of female learning. In the mid-nineteenth century, some scholars mistakenly believed that the title referred not to an actual person but to a technique in writing. The Lettres, with their insightful explanation of the sciences and his core religious and spiritual positions, offer probably the best rounded view of Euler's character” (Calinger, pp. 467-8). Despite being written for the lay reader, these letters broke much new ground: letters 101-108 anticipate the invention of ‘Venn diagrans’ in the 1880s; letter 60 speculates in the existence of what are now called exoplanets orbiting the fixed stars (and whether they might support life), and letters 155-169 describe six methods of determining the longitude of a ship at sea. Rare in commerce, especially in this condition.
In 1741 king Frederick II of Prussia invited Euler to join the Berlin Academy, and he spent the next 25 years there. Frederick asked Euler to be a scientific tutor to his niece, Princess Sophie, who lived in Magdeburg. He accomplished this task by writing, between April 1760 and May 1762, 234 letters in French to the Princess. Throughout his stay in Berlin, Euler wanted to have them put into print, but they remained unpublished in Prussia possibly because of tensions from 1762 onwards between Frederick and Euler, which eventually led to Euler’s departure to Saint Petersburg in June 1766. In Russia the Lettres were published first in French, their language of composition and the second language of Europe. This, together with the popular style and clarity of the Lettres, allowed the work to reach a large readership. The first two volumes of the Lettres appeared in 1768 and the last in 1772.
“The Lettres are a treasure trove from the sciences, a distinctive encyclopedia of knowledge, and they give Euler's mature positions as well as outcomes of earlier research and speculations. He began simply, for his student had little knowledge of natural philosophy or mathematics. But he proceeded quickly to more difficult topics in steps and well-chosen examples lightened with an occasional touch of humor. The text showed his insight, how he worked through problems, and the clarity of his explanations. It contained no mathematics.
“The initial three topics — extension, velocity, and sound — were among the seventy-nine dealing with general physical science that comprised the first of three natural divisions within the Lettres. For René Descartes, extension was the essential property of matter. But Euler assigned two other fundamental properties from the work of Isaac Newton: impenetrability and inertia. He explained these in letters 69, 70 and 74, which make impenetrability the most general property … Euler — like Newton — computed the velocity of sound. For the first edition of his Principia Mathematica Newton had multiplied frequency by wavelength and found the velocity of sound to be 968 English feet per second. Basing his calculation upon new frequency measures of organ pipes, in the second edition in 1713 he increased the figure to 1,020 feet. Euler gave a closer approximation of 1,142 feet per second; the actual value is 1,107. Euler treated light as analogous to sound, as a vibration in the ether. The figure for the speed of sound was minuscule measured against the greatest velocity known, that of light which since the time of Galileo Galilei had not been thought to be instantaneous. Euler calculated that it moved 12,000,000 English miles per minute. That this is not much higher than the actual value of 11,176,943.8 miles per minute could mean that Euler rounded the figure to make its dimension clearer to his student. Letters 4 to 17 proceeded to music, the air, and the atmosphere. Letters 17 to 45 explained light, starting with the systems of Descartes and Newton on the subject, both of which Euler rejected, and proceeded to his pulse theory, optics, the theory of colors, dioptrics, reflection, vision, and the structure of the eye. His opposition to Newton’s corpuscular optics meant not that Euler was anti-Newtonian but that he found fault with Newton's optics on questions of reflection and refraction.
“The ether, seen as an extremely tenuous and elastic form of matter filling in nature all of otherwise empty space, remained the fundamental concept of Euler’s physics. With it he continued to explain most physical phenomena — mechanical, optical, magnetic, and electrical. Ether first appeared in letter 19. Euler’s application of ether to explain celestial motion has been interpreted as Cartesian, but it was not Descartes's ether. Two considerations regarding ether guided Euler. He sought to remove the neo-Cartesian charge that mutual attraction was a secret or occult property, and he rejected Newton's action at a distance in explaining it.
“Letters 45 to 79 addressed gravity and its effects, along with mechanics, cosmology, the tides, and the theory of matter — especially impenetrability. Euler praised Newton for the great discovery of universal gravitation and gave the falling apple example that Voltaire had used. Newton, he wrote, had looked to how the force acting upon the descending apple would be affected if the height of the tree were the distance of the moon. Euler's account helped this example make its way into folklore. A dozen letters discussed attraction, a property with which all celestial bodies are endowed; its effects depended upon mass and proximity. The enormous distances of the stars from us prevent them from affecting the planets in our solar system. Euler accepted attraction over impulsion, for it would not lead to false consequences on such questions as lunar motion and the shape of Earth. Cartesian impulsion required action by contact and rejected action at a distance. Following Bernard le Bovier de Fontenelle on the plurality of worlds, letter 60 found it highly probable that there were inhabitants on other solar planets and what are now called exoplanets around fixed stars. Euler projected an infinite number of the last. Continuing studies of the motion of comets and the moon, he held, had confirmed the exactness of the inverse-square law of mutual gravitational attraction. Letter 61 gave priority to Johann Tobias Mayer for achieving the high degree of precision needed for exactly determining the moon's motion. Euler credited Descartes with identifying the influence of the moon on the tides, a position that he believed the ancients had held. But he rejected Descartes’s idea that the way the moon exercised that influence was to press on the ether as it moves, causing the tides; Euler instead attributed the effect to the moon’s attraction …
“The second section of the Lettres, 80 through 133, inquired into philosophy, religion, logic, and the Euler-Venn diagrams; it covered topics as diverse as liberty, ethics, language, forms of syllogisms, divisibility, monads, and the certainty of scientific, moral, and historical truths. In religion Euler adopted ontological and epistemological arguments against three groups: the Wolffians, the mechanistic materialists, and the idealists. He considered as well the predecessors of idealism, the solipsists represented by Nicolas Malebranche, for whom a person's own mind alone exists and knowledge from outside is unsure. He criticized skeptical freethinkers and the French Encyclopedists, and argued that the world without matter as proposed by the idealists is incomplete, and that God must create Leibniz’s best of all possible worlds. But letter 85 opposes Leibniz’s notion of a pre-established harmony between mind and matter as a fiction leading to a materialistic determinism destructive to liberty. For Euler the connection between mind and matter remained a mystery …
“The most exhaustive and authoritative science popularization written during the eighteenth century, the Lettres critically examined in greater depth than did other works the complex and changing major Enlightenment natural philosophies, the Cartesian, Newtonian, Leibnizian, and Wolffian, and presented each in terms understandable to the educated European reading public. As one of the few leading men of science, well grounded in all four schools of thought and with a sure command of them, Euler could comment adeptly on each. In neither method nor content was Euler a Cartesian, as has been sometimes thought; instead he brought together consistent ideas in the sciences from different natural philosophers and new thought in mathematics and physics introduced by some members of the Bernoulli family, and he combined these with his original insights. This was the Eulerian system. Alexandre Koyré included the Lettres among the prominent Newtonian popularizations — Henry Pemberton’s View of Sir Isaac Newton’s Philosophy, Voltaire's Philosophical Letters and Elements of Newton’s Philosophy, Count Francesco Algarotti’s Newtonianism for the Ladies, Colin Maclaurin’s Account of Sir Isaac Newton’s Philosophical Discoveries, and Pierre-Simon Laplace's subsequent System of the World — but Euler’s work was far greater …
“The third section, letters 134 to 234, was devoted entirely to physical questions. Euler composed this after he visited the future princess in Magdeburg in May 1761, at which time she said she was no longer able to understand his letters completely and asked him to restrict himself to physical questions. Prominent topics here were the nature and causes of electricity and its visible manifestations in sparks from discharges, thunder, and lightning. Euler devoted at least seventeen letters to electricity, 138 - 154, and nineteen to magnetism, 169 - 187. The disequilibrium and elasticity of the ether were crucial to Euler’s explanation of both …
“Letters 155 through 169 began with two of six methods for determining the longitude of a ship, and the narrative here could be seen as a brief history of science. Euler recognized that all six methods had defects and would require corrections. The first depended upon carefully measuring the direction and length of a journey and placing the result on a map to obtain, as he commented, a rough approximation of a position at sea. Euler noted that the currents in the Atlantic Ocean made a voyage from Africa to America take less time than the reverse. His second method for obtaining longitude, the most classical, depended upon a proposed precise timepiece, a recording of the time passed between a single event — such as the sun at noon at a reference point — and the given location. From these Euler computed the angle of rotation of Earth, knowing that it turned fifteen degrees in an hour. The difference between the two points gave the angle of rotation, which provided the difference in longitude. It was only with the accurate timepieces invented by John Harrison in the 1760s that his method became useful. Euler referred to the British Parliament prize competition on the subject of longitude but did not mention Harrison; he wrote these letters in 1761, four years before he was awarded a small portion of that prize.
“Theoretically Euler’s next three methods for finding longitude resembled that making use of the clock, while the last employed a magnetic needle. In place of the sun’s zenith as the reference point, all but the last appealed to some astronomical phenomenon. A third method used eclipses of the moon and eight simultaneous equations. Each eclipse had to be measured at two different places and the findings compared with known longitudes. For the least possible error, the observations had to be combined. Still another method relied on eclipses of the satellites of Jupiter, and a fifth observed the daily motion of the moon and determined its velocity. Each procedure required making tables and comparing the result at the departure point with that for the final location. The lunar method using eclipses carried in practice the highest degree of precision, but it was not possible to ascertain without an error the moon's true place for every moment. Guided by prize competitions of the Paris Academy of Sciences, Euler explained a sixth method in which sailors employed a magnetic needle and its declination to chart their way at sea.
“Letters 101 to 108, written in February and March 1761, introduce what are today known as Venn diagrams, though that is a misnomer. Diagrammatic representations in logic were not original with Euler; they appeared in some eighteenth-century treatises on the subject and it is possible that Johann Heinrich Lambert employed them shortly before Euler's Lettres. In letters 101 and 102 Euler stressed the need for a disciplined language in representing general ideas and expanding upon them; he employed circles in diagrams to explain different forms of syllogisms and hypothetical propositions. The rules of reasoning held that if two propositions in a syllogism were accepted, the third that followed necessarily from them must also be true. Euler explained joint and independent areas of affirmative positive universal, negative universal, affirmative particular, negative particular, and more with two or three intersecting circles. An example is “for every A is B; but no C is B, and no B is C; therefore no C is A.”
“In the article ‘On the Diagrammatic and Mechanical Representation of Propositions and Reasoning’ more than a century later, in 1880, the Cambridge mathematician John Venn accepted only the diagrams in logic that he called “Eulerian circles.” He added ovals for representations and found that the same diagrams could be utilized to analyze different lists of propositions by closely following which compartments were empty. Today the representations may be called Euler-Venn diagrams. Euler further proposed a design for a logic machine, but no record exists that the machine was ever constructed.
“The influence of the Lettres in German culture went beyond the sciences. Among others, Immanuel Kant, Johann Wolfgang von Goethe, and Arthur Schopenhauer praised them. Although Euler was comparatively weak in philosophy, Kant read the Lettres before criticizing Wolffian dogmatic rationalism. Most philosophers agree that Kant’s transcendental idealism, based upon the view that space and time were not abstractions from the physical world, was directly indebted to Euler, as apparently was Kant’s treatment of the impenetrability of atoms … Goethe regarded Euler highly as an original thinker, who with his concept of the achromatic eye had disproved part of Newton's corpuscular optics. In his magnum opus, Die Welt als Wille und Vorstellung (The world as will and representation), which was published as one volume in the first edition (1818) and as two volumes in the second (1844), Schopenhauer — via the application from Kant of the principle of sufficient reason — envisioned an endlessly violent and ultimately irrational world. Still, in chapter 15 of volume 2 of the second edition, he lauded Euler’s Lettres for their insight, precision, and clarity of concepts, presented with a new charm. Schopenhauer likened Euler's achievement “als hatte man ein schlechtes Fernrohr gegen ein gutes vertauscht” (as that of a man who had exchanged a poor telescope for a good one). The Lettres, he wrote, revealed cogently the fundamental truths of mechanics and optics” (Calinger, pp. 461-9).
From the final years of Euler’s life to the present, the Lettres have met with phenomenal success. He probably prepared in Russia the first German translation, while between 1768 and 1774 Euler’s student Stepan Rumovskij translated the Lettres from French into Russian, the first of five Russian editions leading up to 1808. By 1800 they had gone through thirty editions and were translated into seven other languages: Danish, Dutch, English, German, Italian, Spanish, and Swedish.
Calinger, Leonhard Euler: Mathematical Genius in the Enlightenment, 2015; Eneström 343, 344 & 417; Houzeau & Lancaster 8897; PMM 196 (note); Honeyman 1074; DSB IV, 471.
Three vols., 8vo, pp. xii, 314; xiv, 340; xiv, , 404, with twelve folding engraved plates and numerous woodcut diagrams in text. Contemporary plain wrappers, entirely uncut. Wrappers with fraying and some wear.