Leonhard Euler 1707-1783 Emil A. Fellmann, Basle (CH)

Two hundred years after the death of Euler, research into his prolific writings continues to reveal his extraordinary prescience

In honour of by far the most productive titles of publications about Euler (Burck- mathematician and physicist that has ever hardt - Verzeichnis). lived — Leonhard Euler — a memorial edi­ tion of his life and work1), referred to here Table 1 — Main Periods in the Life of Euler as EGB, has been commissioned by his 1707 - 1727 Basle (Switzerland) native city of Basle. Twenty nine specialists 1727 - 1741 St. Petersburg (Russia) from ten nations and four continents have 1741 - 1766 Berlin (Prussia) combined to produce this carefully blended 1766 - 1783 St. Petersburg synthesis of his scientific and literary work. The Opera omnia of Euler, well-known to Mechanics every mathematician and historian of In the introduction to his two-volume science (their covers at least) have been ap­ Mechanica (Petersburg 1736) Euler pre­ pearing since 1911 through an international sents an extensive revue of this science collaboration under the auspices of the with, as main feature, the systematic ap­ the Weierstrass student S.V. Kovalevskaja Swiss Academy of Sciences. They occupy plication of analysis to the then current as (1888). more than 70 quarto volumes and are being well as new problems in mechanics. The In an appendix to his Methodus inve­ supplemented by a fourth series (Series predecessors of Euler had adopted a largely niendi ... (1744), his detailed exposition on quarta) of 14 volumes containing his scien­ synthetic-geometric approach, Newton's the calculus of variations, Euler suggests a tific correspondence (Series IV A, 8 vo­ immortal Principia mathematica serving as formulation for the famous-infamous "prin­ lumes) and note and day books (Series IV a pregnant example. Euler's approach here ciple of least action" for the case of the B, 6 volumes). Two volumes of these series — as later also in optics — was fully motion of a point mass under the influence have been published since 1975: volume IV analytical, applying systematic analytical of a central force : the relevant trajectory A, 1 (1975) offers a global survey of the methods that were to lead to clear and minimizes the integral ∫ mv ds, whereas roughly 3000 known letters to and from direct descriptions and then solutions of Maupertuis, the then President of the Aca­ Euler, and volume IV A, 5 (1980) contains the relevant problems. The full title of the demy of Berlin, at almost the same time, the correspondence of Euler with Clairaut, book expressed the essential theme : Me­ propounded the principle in a much more d'Alembert and Lagrange2). chanics or the Science of Motion, analyti­ special way. The figures alone show Leonhard Euler cally presented. In the second appendix of the Methodus to have been one of the greatest scholars Euler starts with the kinematics and the inveniendi,..., following a suggestion of of all times. A cosmopolitan in the true dynamics of a point mass in vacuum and in Daniel Bernoullis — Euler applied variation sense of the word — he lived his first twen­ a resistant medium. The section dealing calculus to the theory of beam bending and ty years in Basle, was active altogether with the motion of a point mass under the arrived via the relation more than thirty years in Petersburg (now influence of a force directed towards a ∫ ds/R2 = C ∫ y"2 dx / ( 1 + y'2)5/2 Leningrad) and a quarter of a century in fixed point is a brilliant analytical reformula­ where R is the radius of curvature and C is Berlin (then in Prussia) — Euler attained tion of the corresponding chapter in a constant, at the really spectacular "Euler a celebrity and popularity with which but Newton's Principia. In the second volume bending formula" without which engineer­ few scholars, as e.g. Galileo, Newton and he examines the forced motion of a point ing sciences would be unthinkable even to­ Einstein can compare. mass and solves a number of differential- day In the 18th century, mathematics and geometric problems of surface theory and P = π2Ek2/4f2 physics were not yet separate disciplines the theory of geodetic lines. Almost thirty in which Ek2 is the "absolute elasticity" and in the case of Euler it is particularly dif­ years later, Euler gave in the Theoria motus (rigidity) and 2f the length of a bar sup­ ficult to define his field of activity : his com­ (1765) a new presentation of point mecha­ ported at both ends. Besides this first plete publications count "only" about 30 nics in three dimensions, resolving accor­ calculation of an elastostatic eigen value, volumes (in modern terms) of pure mathe­ ding to the model of Maclaurin (1742), the Euler was also the first to calculate the matics; the majority of the remainder are force vectors on to a fixed, orthogonal sys­ elasto-kinetic property of the eigen fre­ spread over physics, astronomy, techni­ tem of coordinates. In addition, in his ana­ quencies of a transversally oscillating ques (i.e. engineering sciences), philoso­ lyses of rotating bodies, he established the beam. phy, theology and music theory. Many of differential equations relating the dynamics his manuscripts which are in Leningrad to the main inertial axis, which characte­ Hydromechanics have not yet been studied thoroughly, nor rizes the motion. He also formulated the In the domain of hydromechanics, the have they been edited. We have still to see law, expressed as an elliptic integral, first big work of Euler was his extensive what lies in store for present day resear­ describing the motion of a rigid body round opus on vessels, the Scientia navalis chers, but it can be expected that not only a fixed point, "Euler's angle", to which he (1749). This work represents after the historians will find them rewarding. In the was led when studying the precession of Mechanica ... the second milestone in the following we shall try to summarise the the equinoxes and the nutation of the development of rational mechanics and it main work of Euler in physics. For mathe­ Earth's axis. Other special cases of the has lost nothing of its importance up to the matics and disciplines not dealt with here, theory of gyration, where the differential present day. Not only, for the first time, as well as Euler's biography see EGB which equations can be solved, were discovered were the principles of hydrostatics set out contains a bibliography of more than 700 and dealt with by Lagrange (1788) and by in splendid clarity and based on it the scien­ 6 tific fundamentals of the theory of ship­ wave theory akin to Huygen's. However, in analytically. Certainly he restricted himself building, but the subjects covered gave an England, opposition to the emission theory in his theory always to points on the axis, overview of almost all relevant lines of was long in coming : with the exception of but in this case he treated aperture errors development in mechanics during the 18th Robert Hooke, the first British physicist of and chromatic errors thoroughly and com­ century (W. Habicht in EGB p. 243). high calibre openly to stand up against it pletely as nobody else; so at least the In the first volume Euler deals with the was Thomas Young in his Bakerian Lecture theory of the astronomical telescope was general theory of the equilibrium of floating — it is true, with the most important argu­ brought to a tentative conclusion. But bodies — at that time a "novum" — pro­ ment which Euler had not yet at his dispo­ Euler was subject to a fundamental and blems of stability as well as small oscilla­ sal : the theory of interference. fatal error in assuming that the aberration tions in the neighbourhood of the state of Subsequent to the utilization of the effects at axis-oblique light-incidence equilibrium. In this connexion he defined, refractor telescope by Galileo and Harriot at (aplanasy and koma errors) could be via the pressure of liquids (independent of the beginning of the 17th century, the then neglected with respect to aperture errors the direction), the "ideal liquid", which, unavoidable colour rings in the picture field (spherical aberration). This is, in practice unquestionably provided Cauchy later on proved to be very disturbing. Because of not the case since all errors are of the same with the pattern for his definition of the this, David Gregory and Newton turned to order of magnitude. Nevertheless the fin­ strain tensor. The second volume gives the the (in this respect) better reflector dings of Euler are astonishing, even if one applications of the general theory to the telescope. Only on the basis of Newton's simply regards — in comparison with the special case of a ship. With the Scientia examination of the dispersion of light in a famous optics of Gauss — his theory esta­ navalis, Euler established a new science prism could the possibility of eliminating blished in 1765 Théorie générale de la diop- and influenced markedly the development the chromatic errors be envisaged. Newton trique (cf. W. Habicht in EGB p. 283). of navigation and marine engineering. Only himself initially considered it impossible to One should not forget that one frequent­ a few specialists are aware of the fact that reach achromasy by employing materials of ly finds in Euler's works remarks which are it is due to none other than Leonhard Euler different refractive index. That was true off the central theme. These have often fer­ that we first understood the technically also at the beginning for the London opti­ tilized or even anticipated the subsequent realisable principles of the impeller drive cian John Dollond, until he was successful work of other scientists. In this respect one and the propeller. The bold projects of (after having followed several misleading can cite the difference between light-power Euler's time for its application were, of paths) in building an achromatic assembly and illumination-power, as one meets later course, condemned to remain theoretical, from a combination of lenses of crown- on in Lambert's Photometria. William since the necessary propulsion energies and flintglass. The contribution of Euler to Herschel, in a certain sense, was continu­ were not yet available. this famous discovery was considerable, ing Euler's dioptric studies when making Other work that is well known in the although one is not allowed to say that calculations for duplets and telescope eye­ history of techniques, are Euler's ex­ Euler had discovered the achromatic pieces. periments on the Segner hydraulic power telescope (cf. EGB p. 310). Dollond was The penetration and richness of Euler in machine and the associated theory of the decisively influenced by a publication all his work including optics remain asto­ water turbine. Just 40 years ago, J. (1749) of Euler as well as a paper by the nishing — Optica occupy seven quarto Ackeret had such a turbine constructed ac­ Swedish Samuel Klingenstjerna, which volumes — and what is almost unbelie­ cording to the specifications of Euler and was stimulated directly by the study of vable, is that his main work on light was determined that its efficiency lay above Euler. the work of a blind man. 71%. This is a sensational result bearing in All that has been said concerning me­ mind that today, employing the most up to thods in mechanics is no less true for the REFERENCES three-volume giant work Dioptrica date techniques, the efficiency of a turbine 1. Leonhard Euler 1707-1783, Beiträge zu Leben of comparable dimensions lies slightly over (1769/71), which was for a long time the und Werk, Gedenkband des Kantons Basel- 80%. textbook on geometric optics and was Stadt (Birkhäuser Verlag, Basle) 1983, 555 pp. The formulation of some truly classic Euler's own Synopsis. In contrast to his 2. Op. cit. pp. 489-509 and the prospectus papers on the analytical theory of fluid predecessors, who almost exclusively pro­ Leonhard Euler - Opera omnia (Birkhäuser mechanics, for which Euler developed a ceeded synthetically, Euler treated optics Verlag, Basle) 1982. system of fundamental formulae on hydro­ statics and hydrodynamics, came in the Table 2 — Main Works of Leonhard Euler early fifties of the 18th century. Among Shortened titles in chronological order according to year of publication them one finds the continuity equation for fluids of constant density, the equation for 1736 Mechanica (2 volumes) velocity potential (normally named after 1738 1740 Rechenkunst (2 volumes) (Arithmetics) Laplace) and the general "Euler equations" 1739 Tentamen novae theoriae musicae (Music theory) for the motion of ideal (i.e. frictionless 1744 Methodus inveniendi (Calculus of variations) flowing) compressible and non-compres- 1744 Theoria motuum planetarum et cometarum (Mechanics of celestial bodies) sible liquids. Characterising this area of 1745 Neue Grundsätze der Artillerie (Ballistics) work is the application of certain partial dif­ 1747 Rettung der göttlichen Offenbarung gegen die Einwürfe der Freygeister (Theology) ferential equations to the relevant pro­ 1748 Introductio in analysin infinitorum (Introduction, 2 volumes) blems. Of these, Euler was especially proud 1749 Scientia navalis (Theory of vessels, 2 volumes) as we know from his own statements — 1753 Theoria motus lunae (First lunar theory) and he had the right to be. 1755 Institutiones calculi differentialis (Differential calculus, 2 volumes) 1762 Constructio lentium objectivarum (Achromatic lenses) Optics 1765 Theoria motus corporum (Second mechanics) Euler was occupied with questions of op­ 1766 Théorie générale de la dioptrique (Theory of lenses) tics during the whole of his life, and no­ 1768 Lettres à une Princesse d'Allemagne (Philosophical letters, 2 volumes) where does one see more evidently than in 1768 Institutiones calculi integralis (Integral calculus, 3 volumes through to 1770) 1769 Dioptrica (Optics, 3 volumes through to 1771) this science the contrast with the school of 1770 Vollständige Anleitung zur Algebra (Algebra, 2 volumes, 1768 preprint of a Russian translation) Newton. In his first papers on optics (Nova 1772 Theoria motuum lunae (Second lunar theory) theoria lucis et colorum, 1746) Euler oppos­ 1773 Théorie complète de la construction et de la manœuvre des vaisseaux (Second theory of ed the corpuscular theory of Newton with a vessels) 7