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: Feb. 3rd , 2023:
The work on the transcription of the conic sections of Appolonius by Gregorius continues now with the parabola, of which there is an introduction and the 8 parts are presented here : prolegomena ; part1 ; part2 ; part3; part4; part5; part6; part7; part8;
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.
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 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.