*General Introduction : The
State of this Site Aug. 2020. *

**This
website is now 14 years old : There are now in excess of 880 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 amounted roughly to a 500 page
book being printed from the website
worldwide every 15 minutes. There was, of course, some seasonal variation
depending on semester demand. However the Corona virus has led to quiet times
and the occasional very busy times, though the current download rate is at most
half of what it was a year ago; it is unfortunate for humanity in general that
this catastrophe has happened.**

**
One of my former colleagues at Adelaide University, Ernest Hirsch,
passed away earlier in 2015, after a long and fruitful life after
arriving in **

**
*** 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. At
present I am concerned with Euler's work on fluid flow, parts I and II of an
essential text book are complete, and work has begun on Part III, which
considers different kinds of flow; prior to this, a treatment of the analysis of continued fractions by Euler
was given, with applications to square root extraction, etc. I have now
completed 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 **,

** ***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 5 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. Nov. 2020.*

*Feel free to contact me for any relevant reason as discussed ;
my email address at present is :*

*ian.bruce@ace.net.au *

**Latest addition
Feb. 28 ^{th} , 2021: **

*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. At present we have the file that we will call Part I: etr258.pdf which
includes some of the basic equations such as that for continuity for ideal
fluids, without the messy contributions due to viscosity and surface tension,
etc., and which are now presented by means of vector analysis. The
following paper taken as Part II : E396 is presented here in two parts for
convenience Chapters 1-3: etr396p1.pdf, and Chapters 4-6 : etr396p2.pdf . Here** **Euler
is concerned more with establishing basic equations rather than solving
individual problems, which were of course fundamental to the further
development of the subject. Horace Lamb, for example, in his initial *Treatise on the Motion of Fluids*
published in 1879, begins his work with the foundations laid by Euler in these
papers; this book incidently was written here in

Part III is now complete. The fluid is taken to be water. Chapter 1 :
etr409ch1.pdf ,
regarding fluid flow in one dimension or in tubes, which uses two fundamental
equations described previously. Chapter 2
: etr409ch2.pdf , considers the flow
along capillary tubes of various geometries. Chapter 3 : etr409ch3pdf , considers the detailed solutions of
problems regarding the flow of water in various kinds of tubes, U tubes, etc.,
and the use of pumps as simple machines in raising water. Chapter 4 etr409ch4pdf , performs
a detailed analysis of the use of individual pumps raising water with the aid
of pistons, initially alone but finally with 2 or 4 pumps working together; the
use of Euler's analysis might be able to be applied to the working of the human
heart considered as 2 pumps working out of phase with each other, as considered
here. Chapter 5 is rather surprising etr409ch5pdf . Here
Euler is concerned with the properties of water flowing inside a closed glass
circular tube set in the vertical plane; heat is supplied at one end of the
horizontal diameter and cooling at the other end, and the laminar flow of water
flowing as a convection current is analysed, making use of the density changes
of water on heating or cooling water; later the locations of heating and
cooling are changed to various angles, including the vertical; in the course of
these experimental observations Euler and his experiments observed the
anomalous expansion of water a few degrees above freezing point, and also the
change of the convection currents from laminar to turbulent flow under certain
conditions.

Part IV. Here the fluid is taken to be air; immediately numerous
difficulties arise, as Euler attempts to use the same two basic equations
developed for liquid flow. At present we offer Chapter I : etr424ch1.pdf ; at this
stage Euler was unable to provide a differential equation describing the motion
of a small packet or particle of air, and made use of scales or orders of
magnitude of the observable effects, such as pressure and density changes
associated with the translations involved. Chapter 2 is now presented in which a sound
pulse or pulses are sent down a hollow tube
etr424ch2.pdf ; still more difficulties arise as Euler considers the
fundamental properties of sound waves, such as may arise as pulses ; some of
the properties of the pulses , both rarefied and compressed, seem to be in
order; other assumptions appear difficult to verify experimentally, and are
derived from his desire to use the basic equations already established, in some
form. Chapter 3 etr424ch3.pdf is an
application of the theory to the notes produced by flutes; recall that Euler
played the flute; it is clear now how Euler's theory is deficient, and although
some properties of air pulses in tubes are shown correctly, the fact that the
pressure and density changes induced are linear is a major deficiency;
essentially the wave equation should be solved, resulting in s.h.m., as we well
know. However, it is interesting to see how someone as bright as Euler could be
led astray; as no doubt modern day scientists are, on investigating the
boundary between what is known to be true, and what is still at least partially
unknown. Particle physics springs to mind here, where we seem to have reached
such an impassible boundary. However, Lagrange was soon on the scene, and
removed most of the difficulties Euler experienced.

**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. *

*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, **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. **E*281 has now been added
to the general Euler papers . *E*279 has now been translated, relating to the general quadratic
being a perfect square to be found below; following on from this, *E*323 has been added, which
provided a new algorithm for solving Pell type equations. In addition, *E*071 , 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 **

*Euler*** : E17 : **

*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** ;

*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.*

*A**n 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. **

**The translation
of the final chapter of Book 3 of Euler's Dioptrics….,
E404, concerning microscopes is now
complete **

**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. **

*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.__**
Feb. 28 ^{th }, 2021, **

*iandotbruce@acedotnetdotau** .*