Euler's early papers mathematical papers show the influence of Johanne Bernoulli, his mentor. E001 and E003 are concerned with the isochrone and brachistochrone problems. E002 is a dissertation on sound, which was presented to the University at Basle in a vain attempt to gain the vacant physics chair; E005 is concerned with curves that intersect orthogonally, a much longer paper, and this is perhaps Euler's first mature paper; all of course show his mathematical brilliance, although he has not yet made the transition to completely analytical methods. E006 is concerned with the involute of the circle, formed by unwinding a string along the tangent, as a correcting mechanism for the period of a chronometer, developed originally by Huygens in the Horologia. E007 is an attempt to explain atmospheric phenomena in terms of air vesicles, fine matter, and centrufugal force. E008 is about the general solution of heavy planar curves under varous loadings, catenaries, sails, etc. E009 is a classic early paper on the shortest curve joining two points on a surface. E010 is the start of Euler's love affair with the exponential function, related to easing the pain of solving differential equations. The papers presented here in pdf format are taken from the appropriate volumes of Euler's works. E011 is a later paper, and relies on previous work not yet covered in this series of translations. E012 is a masterful work, in which Euler first establishes a surprisingly simple geometric condition for tautochromic curves, and then shows how to generate such curves, both analytic and algebraic, starting from the familiar cycloid ; E013 extends the analysis to a resistive medium where the resistance is in proportion to the square of the speed. E014 is an elementary treatment of finding the pole star from three measurements on a star over time. Both the translation and the latin version are presented, the latter usually at the end. These are amongst the most popular of the translations presented on this site. I wish to thank the members of the Euler Archive based at Dartmouth College for providing most of the original pdf files.
CONTENTS
The (erroneous) construction of tautochronous curves in media with different forms of resistance. This is Euler's first paper E001; the math in the translation has been clarified (Feb. '07) to account for Euler's mistake.
Link to E001 by clicking here.
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E002
A short survey in two chapters of the state of the theory of sound at the time and of sources, with special reference to the flute.
E003
A method for finding algebraic reciprocal trajectories is presented, now complete!.
E005
The Solution of the Problem of Reciprocal Trajectories. This problem had been solved initially by the editor of the 3rd edition of Newton's Principia, Pemberton, and the solution send in code to Johan Bernoulli as a challenge. This was a source of great embarrassment to Bernoulli, though of course he came up with his own solution; hence Euler's interest in presenting a general solution.
E006
A blend of analysis, dynamics, geometry, and algebra is used in this interesting paper to derive the curve necessary to provide s.h.m. for large angles for a cylinder which forms the timing mechanism for a marine chronometer, modifying an original design of Sully. The associated correcting curve is the involute of the circle. Both analytic and algebraic solutions are given.
E007
What amounts to an equation of state is established for air, even though temperature enters in an odd way. However, it is based on a hypothetical model due to J. Bernoulli, involving spherical vesicles containing fine matter with a permeable membrane inflated by centrifugal force. Euler's ability shines through as he analyses this model, but he is unable to fit it to experimental data in a convincing manner, which is not really very surprising. However, a read of this paper will convince you of the progress made in the last 300 years in understanding the structure of matter.
E008
Another blend of analysis, statics, geometry, and algebra is used here in this interesting paper, in which the Calculus has definitely 'Come of Age', as it were. Heavy curves associated with chains, ropes, sails, strings, etc, are analysed both with and without elasticity, after a general formula has been found from statics principles.
E009
In which Euler determines a general differential equation by means of which points on surfaces are joined by curves of the shortest length. The method is applied to general cylinders, conical surfaces, and surfaces of revolution.
E010
In which Euler presents a method for reducing the order of certain second order differential equation to first order by means of changes of variables using exponential functions, for which one variable has a constant rate of change; and these reductions can also be established directly from carefully chosen exponential transformations.
E011
In which Euler presents further methods for transforming differential equations; however, it dates from 1735, and lay some years in the future from his current work, and quotes from a paper E031 on Riccati differential equations.
Riccati
In which Count Riccati presents his thoughts on solving second order differential equations by using some cunning transformations that were to inspire Euler. Originally published in the Act. Er. of 1724.
E012
In which Euler presents methods for producing tautochrone curves including cycloids and other curves, after establishing an interesting basic geometrical principle involving a differential of order zero from the height of the curve.
E013 In which Euler presents a method for producing tautochrone curves in a resistive medium. The method compares the motions of both cylinders and spheres on a cycloid in a vacuum with those on the curves sought. In the analysis, the series expansion for the exponential function is introduced, as well as the procedure for evaluating a constant of integration.
E014
In which Euler presents a method for checking the elevation of the pole star and the declination of a fixed star from 3 measurements over a period of time.
E015 E016 E019 E020 E021 E025
A translation of Euler's Mechanica Vol. I is available here.
A translation of Euler's Mechanica Vol. II is available here.
A fascinating paper in which Euler explores transcendental progressions in which the general terms are infinite preducts related to quotients of factorials - relating initially to the Wallis product for pi, and in which integrals are found for the general terms; a derivation of the 'half-derivative' is given finally as a final consequence. This paper is considered as an application of the next paper presented E020. [An error in the original date of publication has been corrected ; Dec. '08]
Another fascinating paper in which Euler explores the summation of transcendental progressions in which the general terms are generally derived from the familiar g.p.; the beginnings of Euler's fascination with the sum of the inverse squares of integers is made here.
Here Euler returns to enlarge on an earlier paper, E006, but using the later approach of E012, and investigates a possible timing mechanism for a clock, in which two weights unwind from curves attached to a pulley. The physics is interesting, as he shows how the vis viva method is used with the rotational motion of the pulley. The latter part of the paper is purely mathematical, and relates to finding appropriate pairs of isochronous curves : one may be given and the other found;or both are the same. A number of quoted results come from the Mechanica without reference.
This completes with E019 and E020 a trilogy of papers on the summation of series. A synthetic method is demonstrated whereby many kinds of series can be summed using integration and differentiation to reduce a sum to a geometric series. This paper culminates on summing hypergeometric series; a truly fascinating work.
E031
This is a most interesting paper, and shows the extent that the genius of Euler had reached at this time; he applies himself to a new method of solving first order differential equations, as applied to the generel Ricatti equation : though it is hard to classify his method. Certainly series expansions of variables are made as power series, integrations are performed as a summation of special series technique, and results are obtained in general as integrals. A small flaw arises when Euler is absent-minded about an integration. I have pointed it out, but left the subsequent work as it is, as the method is the same, just the wrong coefficients. People have pointed out this paper as seminal in the developement of integral transforms, and it is certainly heading in this direction, but it has its origins fixed in traditional methods. Well worth a read , I think. A few mistakes in the original have been amended from the corrected version in the Opera Omnia as noted in the text, since this paper was first posted.
The origins of this paper are as an extension of E30, in which the arc length of an ellipse is required. It is referred to near the end of Ch. II of the Mechanica as an alternative way of solving a tautochrone problem proposed to Euler by Daniel Bernoulli.
E041
This is one of Euler's most celebrated papers, in which he demonstrates formulas such as Pi ^2= sum of the inverse squares of the positive integers, and many more, on equating the sine expansion of a circular arc to the infinite product of the simple factors of the associated multiple arcs.
E044
This is an extensive paper that develops a method for finding a family of curves arising from the constant of integration of dz = Pdx, which is treated as the second variable; the rudiments of partial differentiation are presented, and there is an extensive survey of homogeneous functions centred around what is now know as Euler's Theorem for such functions. The origins of this paper would seem to be Proposition 15 of Vol. 2 of the Mechanica, relating to families of tautochronous curves, where an integration relying on Euler's Theorem is required.
E045
This is an equally extensive paper that continues the development of methods for finding a family of curves arising from the constant of integration of dz = Pdx, which is treated as the second variable. A method is developed for finding the modular equation for the first order equation that is extended to cover a number of cases; this in turn is extended to second and higher orders. The method involves finding suitable functions to integrate, starting from a part of the modular equation that is integrable, so that the whole equation is of this form. This paper is noteworthy in addition as it seems to be the first in which the function notation, albeit in a slightly different form from the modern meaning, is introduce. I have not been able to check all the equations at this stage.
On Euler Angles : E478 E698 Lexell's Paper on Spherical Triangles Lexell's Paper on the Motion of a Rigid Body
In which Euler sets out the general formulae for the translation and rotation of any rigid body. Translated by Johan Sten.
In which Euler presents some results relating to spherical triangles, complementing some work by the Finnish astronomer Lexell. Now with imbedded diagrams. Translated by Johan Sten.
In which a young associate of Euler, A. J. Lexell sets out some new results on spherical geometry. Translated by Johan Sten.
In which Lexell continues Euler's work in the translation of a rigid body. Translated by Johan Sten.