Some Mathematical Works of the 17th & 18th Centuries, including Newton's Principia, Euler's Mechanica, Introductio in Analysin, etc., translated mainly from Latin into English.

Most of the translations are from Latin by Ian Bruce, with some papers by others.

See below for latest additions.

General Introduction : The State of this Site Aug. 2019.

This website is now 13 years old : There are now in excess of 880 URLs.  It is pleasing to note that on a monthly basis it attracts around 10,000 visitors, and in excess of 100,000 hits are made,  and that more than 1,000 files are downloaded on a daily basis to mathematicians and students of mathematics in around 150 countries, of which the U.S. accounts for approximately a quarter or more, 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.

           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 year has seen the completion of  my translation of Vol. 4 of Euler's Introduction to Integral Calculus, published from his posthumous papers. 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. At present I am working my way through Euler's Dioptricae, having completed vol. 1, and most of vol.2 on refracting telescopes, apart for the appendices on reflecting telescopes.  I have finished 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, 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; since then I have been involved rather deeply with the second volume of Euler's Optics, which I have just finished translating. Unfortunately, Euler  was unaware of some of  the developments in lens design, and he had turned his version of lens and telescope design into a mathematical exercise where the components were not tested individually as the analysis progressed, and which finally does not appear to have produced telescopes that worked particularly well; he blamed the artificier for the deficiencies. The matter of who was right  and who was wrong is very complex and confused.

    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 Ch. I , Newton's Axioms and the addition to Book II ch. 1 of the Principia,  and some chapters of Euler's Mechanica I & II  and his Introductio in Analysin, completed this last year, parts of Huygens' Horologium are very popular, some early Euler papers, esp. E025, and various parts of Gregorius and James Gregory's Optica Promota also, and there is of course a constant demand for material on logarithms, and Harriot's book salvaged from his posthumous notes is of interest to browsers. Lately Jim Hanson's work on Napier's Bones and Promptuary have been very popular. This site is unique in that it provides the only translation available into English of a number of important works. Texts are presented with the understanding that they cannot be error-free, and reveal the translator's idiosyncrasies to some extent. Occasionally there are some notes to help you along, especially at the start of the chapter.


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 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 Newton's methods are obscure, and it has pleased me to be able to unravel some of these. The Principia never was, and never will be, an 'easy read'; however, Newton was a man from the new age of mathematical science initiated by Kepler and Galileo, opening a hidden door via calculus, and who ushered in almost single handedly the world of the mathematical analysis of physical phenomena: Newton's view of the world, controversial at the time because of the idea of action at a distance,  is still the one that we accept largely. To our store of translations also has been added Napier's De Arte Logistica, the books I & II of Euler's Introductio in analysin infinititorum, his Algebra, Methodus Inveniendi Lineas Curvas Maximi Minimive Gaudentes, dealing with variational problems of integrals, and some later related papers, E296 & E297,  following Lagrange's brilliant suggestions; in addition I have added Gellebrand's second part of Briggs's Trigonometriae Britannicae. See below for later additions.

    Very occasionally someone send me an e-mail, for which they have to decipher my address,  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. Aug. 2019.

Feel free to contact me for any relevant reason as discussed ; my email address can be extracted from :


Latest addition Oct. 15th , 2019:



A new work has been completed here in part,  Euler's  Dioptrics…., E367; now we have Chapters 1-7 translated, which is essentially the whole of Book I:Ch.1-Ch.7 are now complete.

 Some interesting developments were taking place in optics at this time: namely the construction of achromatic lenses for optical instruments. John Dollond was involved in this, and objected to Euler's theoretical handling of the problem. Euler eventually produced E266 on the achromatic doublet, which I have translated here following the link below, along with Dollond's letter to the Royal Society as an appendage (already in English).

A translation of Book II, Sect. 1 of Euler's Dioptrics…., E386 is now complete, apart from the Appendices.

Book II, Sect. 2, Ch. 1 details the construction of telescopes that may have a compound objective as well as a possible compound eyepiece lens. I am including some extra background material at this stage; Euler's Dioptrics is difficult to follow in parts, due to the algebraic methods used, which are not always correct, as indicated in King's book The History of the Telescope; that does not mean, of course, that the work should be neglected, as has been the case until now.

 The translation of Part 3 of Book 2 is now complete, and we can now present Ch.1, Ch.2 and Ch.3, the former is concerned with producing upright images, by the addition of extra lenses. I have included some notes here. Ch.2 is occupied with terrestrial telescopes:  Euler's work was continually frustrated by the imperfect telescopes his artisans could make from his theoretical calculations; the flaw would appear to arise from his use of a formula for the focal length of the objective which he produced in Ch. I of the first volume, long before the different forms of lens aberration were understood. As it was, up to this point, he had not made use of Dollond's achromatic doublet, but used a separated pair of lenses, which did not produce very good results. In Ch.3, Euler is concerned with adapting his theory of terrestial telescopes to the innovations of John Dollond, where the chromatic aberration is reduced by using two lenses of differing refractive index, the first concave and the latter convex. Finally, some worked examples are set out to ease the burden of understanding the theory, and to make the work more understandable. The appendices on reflecting telescopes remains to be translated.     link here.




 Revised versions of early files are available here for download as revisions are made : at present there is only

 Euler's E002 available, where Paragraph 6 was omitted.

    link here  to access this file.


  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

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

Hermann's Phoronomia (1715)  .  Link to the contents document by clicking 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 & E054 & E134 & Fermat letter to Wallis,  E031, E041, E044, and E045 are present also, some of which are referred to in the Mechanica; 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.  Link to the contents document by clicking 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 .

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. Oct. 15th , 2019, 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 :

iandotbruce@acedotnetdotau .