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
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 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. Some of
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 :
ian.bruce@ace.net.au ;
Latest
addition: Oct. 5th , 2022:
The work on the transcription of the conic
sections of Appolonius by Gregorius continues now with the parabola, of which
there are 18 parts : Part 1 ; Part 2 are presented here initially.
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 pwith 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.
Contents.
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, craig, craig1, craig2, & Nova1 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, 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 . Link to the contents document by
clicking here.
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 :
iandotbruce@acedotnetdotau .