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.
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.
E015 E016
A translation of Euler's Mechanica is under way at present, of which Vol. I is currently available, and a start has been made on Vol.II.
Ch.I and Ch.2a are now complete.
E698
In which Euler presents some results relating to spherical triangles, complementing some work by the Finnish astronomer Lexell. Translated by Johan Sten.
Ian Bruce. June 3, 2008 latest revision. Copyright : I reserve the right to publish this translated work in book form. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for legitimate personal or educational uses. Please feel free to contact me if you wish, especially if you have any relevant comments or concerns about this work.