Berlin: Haude & Spener, 1746; 1750; 1751.
First edition, an outstanding copy uncut in the original boards, of Euler’s three-volume Opuscula varii argumenti (although only the first volume bears this title), a collection of thirteen tracts, all except one published here for the first time. The most important is the first printing of Euler’s major treatise on light, Nova theoria lucis et colorum (pp. 169-244 of Tom. 1). According to Casper Hakfoort the wave-particle duality debate in optics really began with Euler’s publication of this work. “When Leonhard Euler published his treatise ‘Nova theoria lucis et colorum’ (A new theory of light and colours) in 1746, he made a contribution to the medium tradition in physical optics that was without parallel in the eighteenth century. The ‘Nova theoria’ constitutes the most lucid, comprehensive, and systematic medium theory of that century. The significance of Euler’s theory can be gauged partly from the fact that it was so widely discussed. No earlier attempts to provide an alternative to the theories developing within the emission tradition had stimulated pens to the same extent as Euler’s ‘Nova theoria’. It is remarkable that a relatively short treatise published as part of a collection of articles on a range of subjects created such resonances, and all the more so when we compare this reaction with the limited response to Huygens’ Traité [de la lumière, 1690] – a complete book – and with the way in which Johann II Bernoulli’s prize essay was virtually ignored [‘Recherches physiques et géometriques sur la question: comment se fait la propagation de la lumiere, 1736, published 1752].” (Hakfoort, Optics in the Age of Euler (1995), p. 72). The twelve further tracts cover topics in astronomy, magnetism, electricity, light, sound and mathematics. The three volumes of the Opuscula have Enestrom numbers 80, 121 & 561; the individual papers are 86-91, 109, 151-154, 173 & 174.
“Having rejected Newton’s theory of light a decade earlier, [Euler] now presented ‘Nova theoria lucis et colorum’, which gave the clearest and most comprehensive theory in a medium in physical optics during the Enlightenment. The medium is the ether. The treatise appeared in the first volume of Euler’s Opuscula varii argumenti, which also contained six fundamental works on mechanics, astronomical tables, smallest particles of matter, prime numbers, and longitude.Earlier in the eighteenth century, the corpuscular theory in Newton's Opticks had competed with the pulse theory in Christiaan Huygens's Traité de la lumière. For its inability to explain double refraction in Iceland crystals or the heating of bodies by light considered to be a pressure, Newton objected to Huygens's hypothesis involving a medium; he also took issue with the idea that light was an instantaneous motion in a medium. Somewhat surprisingly, Newton’s theory of colors had encountered support as well as opposition in the German states; textbooks by [Christian] Wolff defended and transmitted it, but by the 1740s Huygens’s wave theory of light had been repudiated. Efforts by Johann I Bernoulli and Jean-Jacques d’Ortous de Mairan to produce hybrids of the wave and the corpuscular theory in physical optics had failed.
“Euler’s ‘Nova theoria lucis & colorum’ propounds a pulse theory but begins by supplying a more complete basis for a wave theory than had existed. After identifying points of opposition between Huygens's and Newton s optics, he characteristically and deftly synthesized consistencies from them. For his new theory Euler was sometimes called Huygensian but, like [Gabriel] Cramer, he was not content with how Huygens had developed his optics. Adding to this collection of ideas, original insights gained from his powerful intuition in physics, findings of experiments on sound, and Huygens’s analogy between the propagation of sound in air and that of light in an elastic ether, Euler offered a new optical theory: only when it held that light was a pulse motion in the ether did he accept that analogy. While Huygens explained the propagation of light as a tiny displacement of very elastic ether particles, Euler proposed a disturbance of equilibrium in the distribution of density within a subtle, elastic ethereal fluid. In explaining reflection and refraction Euler rejected pulse fronts, envelopes of secondary spherical pulses, which were central to Huygens. He also argued that Huygens could not account well for colors. Euler explained these within the medium by connecting color to frequency of vibrations, an approach close to Nicolas Malebranche's description of vibrations of pressure. Euler did not accept Newton’s explication of colors in book 2 of the Opticks as propagated by a periodicity or fits of particles. Periodicity, he observed, was also a property of waves. By incorporating frequency into the medium tradition, Euler increased its explanatory range. In treating optics Huygens had employed geometry, and in the Principia Mathematica Newton’s acoustics had geometric wave equations of vibratory motion. Drawing heavily upon data from a growing body of research on sounds and to a degree synthetic in optics, Euler began to transform Newton’s wave equations into algebraic language, introducing differential equations for the phenomenon of light” (Calinger, Leonhard Euler (2016), pp. 258-260)
In ‘Conjectura physica circa propagationem soni ac luminis’ (II, pp. 1-22), Euler further investigates the propagation of light in the aether, comparing it to that of sound in air.
The nature of the aether itself was investigated in ‘Recherches physiques sur la nature des moindres parties de la matière’ (I, pp. 287-300), the first paper Euler delivered to the Berlin Academy (in June 1744, though it was not published until two years later). “The paper praised the microscope for opening the study of the infinite and diverse least parts of matter, which Euler labelled as molecules; the subject, he asserted, belongs to physics. To confirm the existence of molecules he presented Newton’s demonstration that the mass of a body is proportional to its inertia and that all molecules have the same density and specific gravity; inertia, he asserted, replaces the two forces that Leibniz and Wolff had attributed to monads. Euler looked for similarities and differences between total force and Leibniz’s living force. The paper presented a very subtle matter, fluid and elastic, comprising the ether that fills the physical universe; this was a view that differed from that of Newton and was sometimes mistakenly associated with the Cartesian concepts of corpuscles and the plenum rather than Euler’s pursuit of fluid mechanics and the beginnings of field theory. While the divisibility of matter is incomparably great, the paper maintained, it is finite and the least elements of it are indivisible” (Calinger, p. 220).
In ‘De relaxione motus planetarum’ (I, pp. 245-276), Euler investigated the effect of the aether of the motion of the planets. “Euler reiterates that if the aether, a subtle matter that fills all of space, has a resistance, then the period times of the planets and comets and the corresponding eccentricities must become smaller; hence, the resistance of the aether should be very small. He points out that as the speed of a planet or comet is decreased by the resistance of the aether, the planet or comet is drawn nearer to the sun by the sun’s force while its speed increases. But these two effects of the sun’s force should force the planet or comet’s distance from the sun, as well as its periodic time, to decrease” (Enestrom).
A more significant astronomical work is ‘Tabulae astronomicae solis et lunae’ (I, pp. 137-168), which Calinger (p. 488) refers to as Euler’s ‘first lunar theory’, preceding those in his two great textbooks Theoria motus lunae (1753) and Theoria motuum lunae (1772). In 1745 Euler had “been investigating lunar motion, and had made corrections to Gottfried Wilhelm Heinsius’s lunar tables. In December 1745 Euler sent the improved tables to Saint Petersburg and to Heinsius, who had moved from the Petersburg Academy to the University of Leipzig the previous year. In these tables Euler computed with differential equations the perihelion and the inclination (obliquity) of the ecliptic. Applying new statistical methods and physical principles that he devised in 1744-45, along with superior skill in the calculus of trigonometrical functions and newly derived inequalities and improved observations, Euler was able to compute ‘Tabulae astronomicae solis & lunae’; this treatise, with its improvements for lunar motion, was published in 1746 as part of his Opuscula varii argumenti, part I” (Calinger, pp. 230-1). Euler’s more accurate lunar tables made an important contribution to the determination of longitude by the lunar distance method, for which he was awarded a prize of £300 by the Board of Longitude in 1765.
The Opuscula include two papers on mechanics. In ‘De motu corporum in superficiebus mobilibus’ (I, pp. 1-136), Euler applies his newly discovered principle of angular momentum to study “the motion of bodies that are constrained to move on a rigid curve that is either in free or rigid motion. In this current paper, he uses a mathematically complex principle that is general in principle to solve some extremely difficult cases of this problem. For example, he solves the problem when a particle is free to move within a curved tube that is free in space. In addition, he foreshadows his theory of rigid bodies by looking at the case of a particle that is free to move in a tube that is given an assigned motion” (Enestrom). ‘De motu corporum flexibilium’ (III, pp. 88-165) is an important early work on elasticity in which, according to Truesdell, Euler “introduces the now usual approach to problems of this kind” (The rational mechanics of flexible or elastic bodies 1638 – 1788, p. 223).
There are four papers on mathematics in the Opuscula, all on very different topics. ‘De numeris amicabilibus’ (II, pp. 23-107) is one of several papers by Euler on ‘amicable numbers.’ These are pairs of whole numbers such that the sum of the proper divisors (i.e., the divisors other than the number itself) of each number is equal to the other number. The simplest example is 220, 284: the divisors of 220 other than 220 itself are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110 whose sum is 284; and those of 284 are 1, 2, 4, 71 and 142 whose sum is 220. This example was known to the ancient Greeks, but prior to 1700 only two other examples were known. In this paper Euler finds 58 others! He does this by studying the properties of what is now called the Euler sigma function: σ(n) is the sum of all the divisors of a number n (including n itself). Then the conditions for m, n to be amicable are that σ(m) – m = n and σ(n) – n = m, i.e., σ(m) = m + n = σ(n). For further details, see Dunham, ‘Euler’s amicable numbers,’ maa.org/sites/default/files/pdf/upload_library/22/Evans/pp.05-07.pdf
In ‘Demonstratio gemina theorematis Neutoniani’ (II, pp. 108-120), Euler gives a ‘double demonstration’ (one algebraic and one using calculus) of some rules Newton gave in his Arithmetica Universalis (1707) for the sums of powers of the roots of a polynomial. Another proof was given by Colin Maclaurin in his Treatise of Algebra (1748). In ‘Animadversiones in rectificationem ellipsis’ (II, pp. 121-166), Euler gives a description of the circumference of an ellipse by means of infinite series. ‘Nova methodus inveniendi traiectorias reciprocas algebraicas’ (III, pp. 54-87) gives the solution of the now-mysterious problem of ‘reciprocal trajectories’, a calculus problem that involves determining curves that intersect their mirror-reflection at a constant angle (see Sandifer, The Early Mathematics of Leonhard Euler, pp. 6-7 for a detailed explanation).
The only treatise in the Opuscula that had been previously published is ‘Dissertatio de magnete’ (III, pp. 1-47) (originally published in Pièces qui ont remporté le Prix de l'Académie Royale des Sciences, 1748, pp. 63-97). “All matter, Euler argued, is endowed with many pores that offer unimpeded passage to the aether. Bodies such as iron, however, which are susceptible to magnetization, are, he suggested, also endowed with certain pores that are so narrow that only the subtlest part of the aether can penetrate them. The process of magnetization involves a rearrangement of the particles of such bodies so that these narrow pores form continuous channels from one side of the body to the other … The aether, Euler said, was highly elastic. Its normal state of equilibrium would be disturbed in the vicinity of a body endowed with the special magnetic channels he had described, because, as it pressed by virtue of its elasticity on the openings of the channels, only its subtler parts would be able to enter. These would therefore become separated from the general mass of aether, and would afterward, he argued, find some difficulty in insinuating themselves again into such small interstices in the ambient mass as those they had vacated. Hence, when they emerged from the far ends of the channels, they would be unable to mingle at once with the ordinary aether they encountered there, and would instead be deflected back … and thus a perpetual circulation would be established … Like so many others he attributed the orientation of a magnet to its aligning itself with the direction of fluid flow wherever it happened to be; the only real novelty in his presentation was that he spelled out the dynamics of the situation in a more acceptable form than had usually been done …
“Almost incidentally, Euler went on to suggest that the mere presence of a magnetic circulation around the Earth might explain the action of gravity. Since the circulation “is generated by the elastic force of the aether,” he argued, “it is necessary that this elastic force is perceptibly diminished around the Earth.” He demonstrated that, if this diminution were inversely proportional to distance from the centre of the Earth, bodies swimming in the aether around the Earth would be subject to a central force inversely proportional to the square of the distance. With this proved, he argued conversely that since, like the Earth, the sun and the other planets also exert inverse-square gravitational forces on bodies in their vicinity, they too must be surrounded by magnetic circulations of the kind he had described!” (Home, Aepinus’s Essay of the Theory of Electricity and Magnetism, pp. 146-8).
The only philosophical work in the Opuscula is ‘Enodatio quaestionis utrum materiae facultas cogitandi tribui possit necne’ (I, pp. 277-286). “In this work, Euler purports to offer incontrovertible proof, based on principles from the field of mechanics, that material bodies cannot possess the capacity for thought. He reduces the question to a clear and logical syllogism, which leads to an attempted proof of the non-corporeality of the mind. This work is an early salvo in the philosophical dispute over the nature of the mind and its theological consequences” (Enestrom).
Honeyman 1064; Wheeler Gift 366. Enestrom, Euler Archive (eulerarchive.maa.org/docs/translations/enestrom/Enestrom_Index.pdf)
Three vols., 4to (230 x 190 mm), pp. [ii], 300; [ii], 166, with one folding engraved plate; [ii], 165, with 5 folding engraved plates. Original boards, uncut. Some water stains and browning.