General Introduction : The State of this Site June, 2022.
This website is now 16 years old : There are now in excess of 900 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 amounts roughly to a 500 page book being printed from the website worldwide every 15 minutes. There is, of course, some seasonal variation depending on semester demand. However the Covid virus has led to quiet times and the occasional very busy times, though after a drop in downloads last year, the current rate has risen far beyond the usual rate; it is most unfortunate for humanity in general that this catastrophe has happened. It has had a very bad effect on education worldwide, and of course on tertiary education.
One of my former colleagues at Adelaide University, Ernest Hirsch, passed away earlier in 2015, after a long and fruitful life after arriving in Australia on the Dunera during WWII : I am honoured to be able to perpetuate his memory here in several works which he translated from Euler's German, at the age of 93.
The last few months has seen the translation of the chapters of the first volume of Euler's work Scientia Navalis, or the Science of Ships; at present ch's 1 to 6 are presented here. This follows the continued translaton of the works of Gregorius started many years ago now; at present Books 1, 2, 3 & 4 are here presented in complete translation. 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. A number of Euler's works have been completed: fluid flow; prior to this, a treatment of the analysis of continued fractions by Euler was given, with applications to square root extraction, etc. 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. I have translated 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….. If you feel like sending me an email for whatever reason, please do so; however you have to unscramble my email address below.
The most popular files downloaded recently not in order have been Euler's
This 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 10 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.
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. Jan. 2022.
Feel free to contact me for any relevant reason as discussed ; my email address at present is, which you have to cut and paste :
Latest addition: May 17th , 2023:
Some new Euler translations : E27 ; E28
The work on the transcription of the conic sections of Appolonius by Gregorius continues now with the parabola, of which there is an introduction and the 8 parts are presented here : prolegomena ; part1 ; part2 ; part3; part4; part5; part6; part7; part8;
Ch. 1 of Euler's E110: Scientia Navalis I : Naval Science has now been translated: Naval Science I CH.1.pdf ; Here Euler considers the static cases of prisms of various cross-section floating on water. This leads to the number of ways a given prism can float on water, either vertically or longitudinally.
Ch. 2 of Euler's E110: Scientia Navalis I : Naval Science has now been translated: Naval Science I CH.2.pdf ; Here Euler for most of the chapter provides a summary of the principles of dynamics needed to understand the following chapters. He is interested especially in considering the weight of the vessel acting downwards through the centre of gravity, and the upthrust generated by the weight of water displaced, which acts at another point, which is variable, and rotations about 3 principle axes are considered.
Ch. 3 of Euler's E110: Scientia Navalis I : Naval Science has now been translated: Naval Science I CH.3.pdf ; This is a long chapter running to around 80 pages, in which initially Euler establishes the principles by which floating bodies may be disturbed from their resting position; various prisms with different cross sections are discussed; later the discussion turns to ship like curves where again the principles of equilibrium are discussed. This is a challenging chapter, and shows Euler's amazing mathematical powers.
Ch. 4 of Euler's E110: Scientia Navalis I : Naval Science has now been translated: Naval Science I CH.4.pdf ; This chapter outlines five of the main features of a ship that need to be examined carefully: These include the depth of the centre of gravity of the ship, the effects if being rotated about the three principle axes, and the effect of the rudder.
Ch. 5 of Euler's E110: Scientia Navalis I : Naval Science has now been translated: Naval Science I CH.5.pdf ; This chapter provides a great deal of information in anticipation of Ch. 6 : the mathematical means according to Euler are established for calculating the resistance of a number of elementary shapes moving through water; this includes numerous side issues where particular integrals are evaluated. This is a long chapter of 78 pages!
Ch. 6 of Euler's E110: Scientia Navalis I : Naval Science has now been translated: Naval Science I CH.6.pdf ;The mathematical means according to Euler are established for calculating the resistance of a number of elementary shapes moving through water, making use of Newton's rule for the resistance of an object moving straight through a fluid such as water at rest; different kinds of geometrical shapes are taken to represent the bows of the ship, where three curves are taken to delineate the shape; the resistance is found by resolving double integrals over the surface for a given shape, e.g. part of a sphere, cone, triangle, etc. This is another long chapter of 90 odd pages, which has taken some 4 months to translate. Chapter 7 does not seem to add much new, and will be handled more slowly with other projects in hand.
Ch. 7 of Euler's E110: Scientia Navalis I : Naval Science has now been translated: Naval Science I CH.7.pdf This is the final chapter of Euler's first extended work on ships; it is concerned with the motion of bodies floating or ships sailing in water; numerous formulas are developed for expressing the forces acting on ships due to water resistance, wind on sails, use of oars, etc., and the paths taken are calculated; there is a great deal worked out in this chapter from basic principles, according to the state of physical knowledge at the time.
This is the start of a truly
mammoth book running to some 1250 pages. At present only Books I, II, & III
have been translated here. The work received a lukewarm reception at the time
(1647) as Gregorius asserted that he could square the circle, as the title
indicates. However, there is a place for this work in the history of mathematics,
as it was one of the forerunners of the theory of integration, and the natural
logarithm was developed from geometic progressions applied to hyperbolic
segments - though the present work does not extend this far.
The introduction to Conic Sections, the Prolegomena, is now presented here in addition to Books 1, 2, & 3; Gregorius has made a slightly different classification of cones than that of the Ancients, which is presented here and compared with those. In addition, Book 3 of Gregorius' work Quadrature of the Circle is now complete.This book is concerned with the further development of classical Greek geometry applied to circles. It is some time since Books 1 and 2 appeared, and the method of presentation has evolved a little since then, so that the current method of presentation has been adopted.
book1: Proportions between line segments ; book2: Geometrical Progressions ;
book3: Circles ; Prolegomena; book4: ellipsepart1to6
Ch. 6 of Euler's De Motu Aeris in Tubis : Concerning the Minimal Motion of Air in Conoidal Tubes has now been translated: etr424ch6.pdf ; this is the final chapter in this series, and involves the comparison of conoidal and cylindrical tubes.
Euler spent some time showing how to produce theorems relating to the expansion of trigonometric functions of some multiple of an angle raised to some power as series involving simple sines and cosines of angles, such as in e246.pdf presented here. In addition we now have e061.pdf, in which a new method is found for expanding the product of p with the sines and cosines of any angles as infinte series of the powers of the reciprocals of whole numbers .
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.
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, E23, E24, 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.
An 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.
Euler's Ideal Fluids:
Euler 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. 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. May. 2nd , 2022, 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 :