General Introduction : The State of this Site Aug. 2020.
This website is now 14 years old : There are now in excess of 880 URLs. It is pleasing to note that at the best of times on a monthly basis it attracted around 10,000 visitors, and in excess of 100,000 hits were made, and that more than 1,000 files were downloaded on a daily basis to mathematicians and students of mathematics in around 150 countries, of which the U.S. accounts for approximately half, on a regular basis. This amounted roughly to a 500 page book being printed from the website worldwide every 15 minutes. There was, of course, some seasonal variation depending on semester demand. However the Corona virus has led to quiet times and the occasional very busy times, though the current download rate is at most half of what it was a year ago; it is unfortunate for humanity in general that this catastrophe has happened.
One of my former colleagues at Adelaide University, Ernest Hirsch,
passed away earlier in 2015, after a long and fruitful life after
The last year has seen the completion of my translation of Euler's Dioptricae: vol. 1 on general principles, vol.2, on refracting and reflecting telescopes, and vol. 3 concerning microscopes. At present I am concerned with Euler's work on fluid flow, parts I and II of an essential text book are complete, and work has begun on Part III, which considers different kinds of flow; prior to this, a treatment of the analysis of continued fractions by Euler was given, with applications to square root extraction, etc. I have now completed Euler's Opuscula Analytica, the last text Euler completed while alive, and in which he wished to draw attention to certain matters he considered noteworthy. I had finished previously Lagrange's Traité de la Resolution des Équationes Numériques de tous les Degrés , in which he tackles the resolution of numerical equations of any degree. A few interesting and relevant commentaries from the time including one by Vandermonde have been introduced, from the 1770's, when attempts were being made to solve 5th order equations. In addition I have translated E17, which is really an introduction to the theory behind arithmetical calculations; the applications are presented a little later, relating to business calculations , exchange rates and so on. This I have not translated so far. I have recently finished E33, which is a tentative mathematical theory of music, produced by Euler in his early days.
A number of authors both of books and papers have made reference to this website, all of whom I would like to thank for their favorable mentions. Occasionally people ask me about actual books of the translated material: none are available from me at present, and the free translation message at the top of each page is an attempt to stop others from attempting the same business, without doing any of the work; occasionally somebody writes to tell me how much they enjoy the mathematics presented here, others have ideas about what I should translate next. The fact that this website is so popular and useful is my only reward, and I hope to continue my translations for a few more years…..
The most popular files downloaded
recently not in order have been Euler's Integration
site is produced, funded, and managed by myself, Dr. Ian Bruce, now an
independent researcher or should I say mathematical hobbyist, whose aim is to
provide the modern mathematical reader with a snapshot of that wonderful
period, from roughly the year 1600 to 1750 or so, when modern analytical
methods came into being, and an understanding of the physical world was
produced hand-in-hand with this development. The work is an ongoing process :
translations of Euler's Mechanica , and his Tractus de Motu Corporum
Rigidorum.....are given, as well as his integral and differential calculus
textbooks and his Introductio in Analysin…. and Methodus
Inveniendi Lineas Curvas Maximi Minimive Gaudentes. Work on Newton's
Principia has been completed some 5 years now ; this includes notes by the
Jesuit Brothers Leseur & Jacquier from their annotated edition, and by myself,
as well as ideas from the books by Chandrasekhar, Brougham & Rouse, etc .
The traditional translates of the
Principia do not give extensive notes, if any at all. Some of
Very occasionally someone send me an e-mail, for which they have to decipher my address so constructed to avoid tedious junk mail, concerning things they are not happy about in the text, and their suggestions may be put in place, if I consider that they have a point. If you feel that there is something wrong somewhere, or if you think that further clarification on some point can be provided, please get in touch via the e-mail link below. On the other hand, if you are pleased with the translations, feel free to tell me so. The amount of labour spent on a given translation suffers from the law of diminishing returns, i.e. more and more has to be done in revision to extract fewer and fewer errors. Happy browsing! IAN BRUCE. Nov. 2020.
Feel free to contact me for any relevant reason as discussed ; my email address at present is :
Latest addition Feb. 28th , 2021:
Euler's Ideal Fluids:
spent some time occasionally investigating the theory of ideal fluids, at a
time when the physical properties of liquids such as viscosity and surface tension
were not fully understood; such ideal fluids were ideally suited to a calculus
based investigation. At present we have the file that we will call Part I: etr258.pdf which
includes some of the basic equations such as that for continuity for ideal
fluids, without the messy contributions due to viscosity and surface tension,
etc., and which are now presented by means of vector analysis. The
following paper taken as Part II : E396 is presented here in two parts for
convenience Chapters 1-3: etr396p1.pdf, and Chapters 4-6 : etr396p2.pdf . Here Euler
is concerned more with establishing basic equations rather than solving
individual problems, which were of course fundamental to the further
development of the subject. Horace Lamb, for example, in his initial Treatise on the Motion of Fluids
published in 1879, begins his work with the foundations laid by Euler in these
papers; this book incidently was written here in
Part III is now complete. The fluid is taken to be water. Chapter 1 : etr409ch1.pdf , regarding fluid flow in one dimension or in tubes, which uses two fundamental equations described previously. Chapter 2 : etr409ch2.pdf , considers the flow along capillary tubes of various geometries. Chapter 3 : etr409ch3pdf , considers the detailed solutions of problems regarding the flow of water in various kinds of tubes, U tubes, etc., and the use of pumps as simple machines in raising water. Chapter 4 etr409ch4pdf , performs a detailed analysis of the use of individual pumps raising water with the aid of pistons, initially alone but finally with 2 or 4 pumps working together; the use of Euler's analysis might be able to be applied to the working of the human heart considered as 2 pumps working out of phase with each other, as considered here. Chapter 5 is rather surprising etr409ch5pdf . Here Euler is concerned with the properties of water flowing inside a closed glass circular tube set in the vertical plane; heat is supplied at one end of the horizontal diameter and cooling at the other end, and the laminar flow of water flowing as a convection current is analysed, making use of the density changes of water on heating or cooling water; later the locations of heating and cooling are changed to various angles, including the vertical; in the course of these experimental observations Euler and his experiments observed the anomalous expansion of water a few degrees above freezing point, and also the change of the convection currents from laminar to turbulent flow under certain conditions.
Part IV. Here the fluid is taken to be air; immediately numerous difficulties arise, as Euler attempts to use the same two basic equations developed for liquid flow. At present we offer Chapter I : etr424ch1.pdf ; at this stage Euler was unable to provide a differential equation describing the motion of a small packet or particle of air, and made use of scales or orders of magnitude of the observable effects, such as pressure and density changes associated with the translations involved. Chapter 2 is now presented in which a sound pulse or pulses are sent down a hollow tube etr424ch2.pdf ; still more difficulties arise as Euler considers the fundamental properties of sound waves, such as may arise as pulses ; some of the properties of the pulses , both rarefied and compressed, seem to be in order; other assumptions appear difficult to verify experimentally, and are derived from his desire to use the basic equations already established, in some form. Chapter 3 etr424ch3.pdf is an application of the theory to the notes produced by flutes; recall that Euler played the flute; it is clear now how Euler's theory is deficient, and although some properties of air pulses in tubes are shown correctly, the fact that the pressure and density changes induced are linear is a major deficiency; essentially the wave equation should be solved, resulting in s.h.m., as we well know. However, it is interesting to see how someone as bright as Euler could be led astray; as no doubt modern day scientists are, on investigating the boundary between what is known to be true, and what is still at least partially unknown. Particle physics springs to mind here, where we seem to have reached such an impassible boundary. However, Lagrange was soon on the scene, and removed most of the difficulties Euler experienced.
Recent Euler works such as the recent Optics are now to be accessed from the Euler works below.
Lagrange Work: 'Traité de la Resolution des Équations Numériques de tous les Degrés' is available now complete. Including Notes I-XIV; E30, E282, and Vandermonde's Resolution of Equations are presented: link here
Mirifici Logarithmorum Canon Descriptio..... (1614), by John Napier. This seminal work by Napier introduced the mathematical world to the wonders of logarithms, and all in a small book of tables. Most of the book, apart from the actual tables, is a manual for solving plane and spherical triangles using logarithms. Included are some interesting identities due to Napier. Jim Hanson's work on Napier's Promptuary and Bones is in place here, with a few other items in the Napier index; note by R. Burn; Link to the contents document by clicking here. You may need to refresh your browser as some files have been amended.
Mirifici Logarithmorum Canon Constructio... (1617); A posthumous work by John Napier. This book along with the above, started a revolution in computing by logarithms. The book is a 'must read' for any serious student of mathematics, young or old. Link to the contents document by clicking here.
De Arte Logistica (1617); A posthumous work by John Napier published by descendent Mark Napier, in 1839. This book sets out the rules for elementary arithmetic and algebra: the first book also presents an interesting introduction to the method of extracting roots of any order, using a fore-runner of what we now call Pascal's Triangle. The second and third books are now also complete. Link to the contents document by clicking here.
Arithmetica Logarithmica, (1624), Henry Briggs. The theory and practice of base 10 logarithms is presented for the first time by Briggs. Link to the contents document by clicking here.
Trigonometria Britannica, (1631), Henry Briggs. The methods used for producing a set of tables for the sine, tangent, and secant together with their logarithms is presented here. The second part, by Henry Gellebrand, is concerned with solving triangles, both planar and spherical. Latin text provided in Gellebrand's sections only. Link to the contents document by clicking here.
Angulares Sectiones, (1617), Francisco Vieta. Edited and presented by Alexander Anderson. Vieta's fundamental work on working out the relations between the sine of an angle and the sine of multiples of the angle is set out in a laborious manner. No Latin text provided. Link to the document by clicking here. It is 25 pages long!
Artis Analyticae Praxis, (1631), 'from the posthumous notes of the philosopher and mathematician Thomas Harriot' , (edited by Walter Warner and others, though no name appears as the author), ' the whole described with care and diligence.' The almost trivial manner in which symbolic algebra was introduced into the mathematical scheme of things is still a cause for some wonder; it had of course been around in a more intuitive form for a long time prior to this publication. Link to the contents document by clicking here.
Optica Promota, (1663), James Gregory. Herein the theory of the first reflecting telescope and a whole theory for elliptic and hyperbolic lenses and mirrors is presented from a geometrical viewpoint. Link to the contents document by clicking here.
Opus Geometricum quadraturae circuli, Gregorius a St. Vincentio, (1647) (Books I & II only at present). A great march via geometric progressions expressed geometrically is undertaken by Gregorius as he examines the idea of a limit, refuting Zeno's Paradox; moving on eventually to discovering the logarithmic property of the hyperbola, before stumbling on the squaring of the circle. This is a long term project! Link to the contents document by clicking here.
A start is made here to translating Leibniz's papers that introduced differential calculus to the world, by means of an extended series of articles in the Acta Eruditorum (AE). At present AE1, AE3, AE3a, AE4, AE5 AE6, AE7, AE8, AE9, AE10, AE11, AE13, AE14, AE18 & AE19 are available; Link to the contents document by clicking here.
Some Euler Papers solving problems relating to isochronous and brachistochrone curves are presented in E001 and E003; a dissertation on sound in E002; Euler's essay on the location and height of masts on ships E004; while reciprocal trajectories are considered in E005 (1729); E006 relates to an application of an isochronous curve; E007 is an essay on air-related phenomena; E008 figures out catenaries and other heavy plane curves; E009 is concerned with the shortest distance between two points on a convex surface; E010 introduces the exponential function as an integrating tool for reducing the order of differential equations; E011 is out of sequence, concerns transformations of differential equations; Ricatti's 1724 paper on second order differential equations is inserted here; E012 & E013 are concerned with tautochrones without & with resistance; E014 is an astronomical calculation; all due to Leonard Euler. E019, E020, E21, E22, E025, E026, E036 E054, & E134 & Fermat letter to Wallis, E031, E041, E044, and E045 are present also, some of which are referred to in the Mechanica;E279 concerned with a general quadratic being a perfect square; E071, E281concerning continued fractions and E323, also E736. Also papers by Lexell and Euler tr. by J. Sten appear here incl. E407 recently, and translations of E524, E842 & E81 by E. Hirsch. Lately I have translated Euler's contributions to the theory of sound: E305, E306, E248 & E307 are now available. E281 has now been added to the general Euler papers . E279 has now been translated, relating to the general quadratic being a perfect square to be found below; following on from this, E323 has been added, which provided a new algorithm for solving Pell type equations. In addition, E071 , Euler's incredible paper on continued fractions, has just been finished , and can be found. I have just finished E036, which is concerned with an exposition of the Chinese Remainder Theorem, where a number is to be found from the remainders given by certain divisors. Two papers on fluids are now included, E258 and part 1 of E396 .
Euler: E33 : A Tentative Exposition of a new Theory of Music…., the whole work, Ch.1-Ch.14, is now complete. link here.
Euler : E17 : The complete Ch.1- Ch.9 of Euler's arithmetic text are available for download. link here
My translation of E015, Book I of Euler's Mechanica has been completed. This was Euler's first major work running to some 500 pages in the original, and included many of his innovative ideas on analysis. This is a complete translation of one of Euler's most important books. Link to the contents document by clicking here.
My translation of E016, Book 2 of Euler's Mechanica has also been completed; this is an even longer text than the above. Both texts give a wonderful insight into Euler's methods, which define the modern approach to analytical mechanics, in spite of a lack of a proper understanding at the time of the conservation laws on which mechanics is grounded. Link to the contents document by clicking here.
The translation of Euler's next major contribution to mechanics is now complete (E289); this contains the first definition of the moment of inertia of a body, and also develops the mathematics of adding infinitesimal velocities about principal axes: Theoria Motus Corporum Solidorum seu Rigida. Link to the contents document by clicking here.
A translation of Euler's Foundations of Integral Calculus now has volumes I, II, III, & IV complete. Supplements 1, 2, comprising E670, 3a is E421, 3b is E463, 3c, E321 ; 4a, 4b; 5a, 5b, 5c, 5d & 5e; 6 &7, comprising E59, E588 & E589 ; E675 & E640 ; E595; E391; E581; Supp. 8a & 8b, E506 & E676; Supp. 9 & 10, E677, E678, E679, E680, and E681, while Supp. 11 consisting of E420, are presented in Vol. IV, attached to the end of Vol. III. You can access these by clicking: Link to volume I or Link to volume II , or Link to volumes III & IV
A translation of Euler's Foundations of Differential Calculus is now complete. You can access these by clicking: Link to DifferentialCalculus .
A translation of Euler's Introduction to Infinite Analysis is now complete with Appendices 1-6 on the nature of surfaces. You can access all of Volumes I and 2 by clicking: Link to Analysis Intro .
A translation of Euler's Methodus Inveniendi Lineas Curvas Maximi Minimive Gaudentes……… is now complete, i.e. the Foundations of the Calculus of Variations, and includes E296 & E297, which explain rather fully the changed view adopted by Euler. You can access it by clicking: Link toMaxMin.
A translation of Euler's translation of Robins' work on gunnery, with remarks, Neue Gründsatze der Artellerie , has now completed; including E853, which is of some interest. You can access it by clicking: Link to Neue Gründsatze.
n early translation of Euler's Letters to a German Princess E343, is presented here in mostly subject bundles. These 233 little essays give a rare insight into Euler's mind, and to the state of physics in the 1760's. Link to the contents vol.1 document by clicking here.
Link to the contents vol.2 document by clicking here.
The translation of Euler's ALGEBRA is now complete ; Link to the contents here .
The translation of Euler's Opuscula Analytica Vol. I is now complete being E550 to E562 inclusive, together with E19 and E122 the sections of Vol. II E586, E587, E588&9, E590, E591, E783, E592, E595 [E594 is already present as Supp. 5e in Vol. IV of the Integral calculus] , E596, E597, &E598, E599, & E600 are also presented in the same contents folder as a direct follow-on. Link to the contents here .
A complete translation of Books I & II of Euler's Dioptrics…., E367 & E386 is now provided here, including the Appendices. link here.
The translation of the final chapter of Book 3 of Euler's Dioptrics…., E404, concerning microscopes is now complete . link here.
Having finished with Euler's Optics for the time being, it seemed to be a good idea to present an English language version of Gauss's famous paper, which established the beginning of modern lens optics, his Dioptrische Untersuchungen or Optical Investigations. link here.
My new translation of Newton's Principia is now complete; this translation includes resetting of all the original type, new diagrams, and additional notes from several sources; an earlier annotated translation of Section VIII of Book II of Newton's Principia on sound is now included in the main flow of the text, which helps in understanding Euler's work De Sono. Link to the contents document by clicking here.
An annotated translation of Johan. Bernoulli's Vibrations of Chords is presented. Link to the contents document by clicking here.
A new translation of Daniel Bernoulli's Hydrodynamicae is now complete. Link to the contents document by clicking here.
An annotated translation of Christian Huygens' Pendulum Clock is presented. Here you will also find the first work by Huygens on the probability of games of chance: De Ratiociniis in Ludo ALeae. Link to the contents document by clicking here.
An annotated translation of Brook Taylor's Methodus Incrementorum Directa & Inversa is presented. Link to the contents document by clicking here.
The Lunes of Hippocratus are extended by Wallenius in a much neglected paper presented 'pro gradu' in 1766 at the Royal Academy of Abo (Turku, in Finland); the student defending the paper was Daniel Wijnquist; a full geometrical derivation of each lune is given, followed by a trigonometric analysis. I wish to thank Johan Sten for drawing my attention to this work, and for his help in tracking down an odd reference. Link to the document by clicking here.
Ian Bruce. Feb. 28th , 2021, latest revision. Copyright : I reserve the right to publish any translated work presented here in book form. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of a work for legitimate private personal or educational uses. Any other form of distribution is illegal and has not been permitted by the author, Ian Bruce, who asserts that the contents of this website are his intellectual property: You are NOT given the right to sell items from this website on the web or otherwise offer 'free' to download in any shape or form, such as e-books. Be very wary of websites that make such offers, there are a few around, usually trying to make money out of someone else's work! This website presents a genuine attempt to provide much mathematics lost to the modern reader, who is unskilled in Latin. Please acknowledge me or this website if you intend to refer to any part of these translations in a journal publication or in a book. Feel free to contact me for any relevant reason as discussed above ; my email address can be extracted from :