With apologies to Samuel Johnson, and not to be taken too seriously : 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 mathematics, 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. 2016: Annual Report.
This
website is now 10 years old plus : I can only say, How time flies when you're
having fun! It is pleasing to note that it attracts around 3500 visits, and
50,000 hits are made, on a monthly basis, and that around 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. There is, of course, some seasonal variation
depending on semester demand.
One of my former colleagues at Adelaide
University, Ernst Hirch, passed away earlier in 2015, after a long and
fruitful life : I am honoured to be able to perpetuate his memory here, and the
chapters of Euler's E842, which he translated from Euler's German, at the age
of 93, are downloaded on a regular basis.
Hermann's
Phoronomia has now been moved into the general
scheme of things Also, Euler's Algebra is now on this site
; Euler can make even the most elementary of mathematics interesting; which should be useful for those who use the
site as a teaching aid. The present translations involve the beginnings of
Lagrange's Works, and also of Vol. 4 of Euler's Introduction to Integral
Calculus, published from his posthumous papers.
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
PREFACE
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 3 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
Occasionally people
send e-mails 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 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. Happy
browsing! IAN BRUCE. Sept. 2016.
Latest addition July 25th, 2017: A new work is started here Opuscula Analytica, Volume I. At present chapters I –VII, corresponding to E550, E551, E552, E553, E554, E555 and E556 respectively; However, in addition I have inserted here so far two articles relevant to some of the derivations, past and present, namely E19 and E122; E551 is quite simple and delightful, and one may take from it the lesson, beware of induction, unless it has been done properly. E552 is one of Euler's best known contributions to the development of the theory of numbers, and marks the beginning of the law of reciprocality for square numbers divided by primes. E553 is concerned with changing the form of continued fractions in a certain way; E554 is a classic work of Euler, in which he investigates the remainders or residues left on dividing the squares and higher powers of the natural numbers by some prime number, usually of a given form. E555 is an interesting study, in which the process of interpolation is extended from a few values of some variable x and a function y to infinitely many values; the process has applications in the determiation of π, and in the summation of infinite series from integrals. E556 is concerned with integral extensions to the basic idea of Pythagorian Triplets, involving a more general formula of the form ax2 +by2 = g z2 .See immediately below for the link.
Previously to this all the parts of Supplements 1 to 11 of Volume IV of Euler's Integral Calculus have been presented in translation on this site ; this is a posthumous work published in 1845 from a number of Euler's papers. See below for further details. A new series of translations of the Works of Lagrange has began but now delayed. At present only the first and second articles of Book I of the collected works have been completed; a start has been made on article 3 on the speed of sound, of which the introduction and Ch. 1 & 2 are presented; I believe that the criticism of the start of Newton's exposition in Part II, Ch. 8 of the Principia by Lagrange is unwarranted, as initially Newton is not discussing sound waves, but gravity waves in the atmosphere (due to changing densities of the air essentially, not to be confused with gravitational waves), which has been mistranslated by everyone until now; see immediately below for the link. Chapter 15 of Part 2 of Book 2 of Euler's Algebra, and the final chapter in this work is presented : this chapter involves the use of cubic equations, and in fact includes Euler's proof of Fermat's Last Theorem for the case of cubes ; see the relevant Euler section below for the link. Hermann's Phoronomia has now finally been completed also. E22 on the collision of bodies added recently also; see Euler below for the link. Previous to this, a series of translations had been undertaking involving Leibniz’s calculus, etc., etc. Otherwise, read on as you wish through the contents, which I intend to tidy up sometime soon, if I can find the time and the energy. As it is, like Topsy, it has ‘just grow'd’.
Contents.
The translation of
Euler's Opuscula Analytica has started ; at present
Ch's. I, II and III have been translated ; being E550,
E551, E552, E553, E554, E555 and E556 together with E19 and
E122 ; Link
to the contents 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 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
.
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.
July 25th , 2017, 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! 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 if you wish (by
placing the mouse pointer on my name here Ian Bruce., when my email address will appear at the
bottom of the page; you then use your own mailer to contact me) , especially if
you have any relevant comments or concerns.