Some Mathematical Works of the 17th & 18th Centuries 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.

With apologies to Samuel Johnson: It is the fate of those who toil at the lower employments of life, to be rather driven by the fear of evil, than attracted by the prospect of good ; to be exposed to censure, without hope of praise; to be disgraced by miscarriage, or punished by neglect, where success would have been without applause, and diligence without reward. Among these unhappy mortals is the translator of Latin mathematical works of days gone by; whom mankind have considered,  not as the pupil, but the slave of science, the pioneer of literature, doomed only to remove rubbish and clear obstructions from the paths through which Learning and Genius press forward to conquest and glory, without a smile on the humble drudge that facilitates their progress. Every other author may aspire to praise; the translator can only hope to escape reproach, and even this negative recompense has been granted to a very few....

General Introduction : The State of this Site Sept. 2012.

However, not withstanding the similarities of the present task with Johnson's remarks about compiling his dictionary, it is pleasing to note that for this website, around 3500 visits and 50,000 hits are made on a monthly basis, and that around 25,000 files are downloaded monthly 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. There is, of course, some seasonal variation depending on semester demand.  The most popular files downloaded recently not in order have been Euler's Integration Ch. I , and some chapters of the Mechanica I & II , parts of Huygens' Horologium are very popular, some early Euler papers, esp. E025, and various parts of Gregorius and James Gregory's Optica Promota, 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. Usually there are some notes to help you along, especially at the start.


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. Work on Newton's Principia is now completed; 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 Galilio, 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, and the first book of Euler's Introductio in analysin infinititorum, while work is almost finished on the appendices to the second book.

    Occasionally people send e-mails concerning things they are not happy about in the text, and their suggestions may be put in place, if 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 hyperlink. 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. The site is now 6 years old!

Happy browsing! IAN BRUCE. August 2012.

Latest addition May 3rd, 2013: I have started on a translation of one of Euler's works E65 on the max. & min. curves initially relating to various simple conditions, and extending to calculus of variations type problems, of which this work is a forerunner ; at present Ch. 1,  Ch. 2 & Ch. 3 of Methodus Inveniendi Lineas Curvas Maximi Minimive Gaudentes……… have been prepared; see below; previous to this, I have been busy with Euler's papers (E305) , (E306), (E307) on the propagation of sound ; see below. Previous to this I have put in place a translation of Euler's paper E524 relating to spherical triangles, which in a very short and straight-forwards manner derives all the formulas relating to spherical triangles, and is well worth reading if your knowledge is hazy or lacking in such matters ; in addition, the second part of the Trigonometria Britannica by Henry Gellibrand has now been translated and presented here; concerning the solution of plane and spherical triangles by using logarithms, essentially by using Napier's circular parts. In addition, a new online version of Euler's classic work Introductio in Analysin Infinitorum is presented here : see below. This completes the main text books of Euler on Calculus and Introductory Analysis. Prior to doing this, I had finished  translating a long forgotten work of John Napier : De Arte Logistica…….


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  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 practise 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 labourous 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.

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, 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,  & 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 is now complete. You can access these by clicking:   Link to volume I   or  Link to volume II  , or  Link to volume III.


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 underway. At present you can access Ch. 1, Ch.2a & 2b, Ch. 3  by clicking:   Link toMaxMin.

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.



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.


An annotated translation of Christian Huygens' Pendulum Clock is presented. 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 3rd , 2013, 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 personal or educational uses. Please feel free to contact me if you wish by clicking on my name here Ian Bruce., especially if you have any relevant comments or concerns.