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....
Introduction.
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 30,000 hits are
made on a monthly basis, and that around 25,000 files are downloaded monthly to
mathematicians and students of mathematics in around 120 countries, of which
the U.S. accounts for approximately half, 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 Dr. Ian Bruce, an independent researcher,
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. At present work on Newton's Principia is under way, and Book I is
almost completed; this includes notes by the Jesuit brothers Leseur &
Janquier 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 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 largely that we accept.
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 5
years old!
Happy browsing! IAN BRUCE. September 2011.
Latest
addition May 17th, 2012: I have decided
to produce an online version of Euler's classic work Introductio
ad Analysin Infinitorum, the first two chapters of vol.1 are now available
: see below. The main reason for doing this new independent translation being
to make this work available to a worldwide audience, free of charge. Prior to
that, I have finished translating a long
forgotten work of John Napier : De Arte Logistica, starting from an instruction
manual on arithmetic, leading on to working with surds and then writing on
algebra, as far as indicating how to solve quadratic and cubic equations
etc., by completing the square, cube,
etc, for higher powers. Before that, my
translation of the 3rd edition of Newton's
Principia was finished, with notes added
according to Leseur & Jacquier from the 'Jesuit edition', and from the work
by Brougham & Routh, especially in Book II, as well as from other
commentators, especially Chandrasekhar in Book III; all the diagrams have been
redrawn Prior to that, some of Euler's
work on the theory of numbers has been presented : E134, E026 & E054 together [they deal mainly
with Fermat's Little Theorem] with a letter in translation sent by Fermat to
Wallis in 1658, which is interesting, as it shows how Fermat regarded his
excursions into the properties of numbers, and Euler's later reaction to his
methods. Euler's Institutiones Calculi Differentialis, The
Foundations of Differential Calculus, has now been translated in full; the three volumes of Euler's Institutiones Calculi Integralis, on the Foundations of
Integral Calculus have been translated
also, corresponding in total to volumes 10,
11, 12 and 13 in the Opera Omnia Series I of Euler's works. This is the first
time a complete translation of these works on the calculus, which laid the
foundations for the future development of mathematical analysis at the close of
the 18th Century, has
appeared in English and may still be of interest to historians of mathematics.
However, do not expect a light read, as there is in excess of 2000 pages!
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.
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. No Latin text provided. 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 E842 & E81 by E. Hirsch. 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 the Analysis of the Infinites is now underway. You can access
chapters 1&2 by clicking: Link to Analysis Intro .
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.
A 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 17th , 2012, latest revision. Copyright : I reserve the right to publish any
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 by clicking on
my name here, especially if you have any relevant comments or concerns.