*General Introduction : The State of this Site May. 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 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 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 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 **,

* 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 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
are :*

*ian.bruce@ace.net.au
;*

**Latest
addition: May 2**^{nd} , 2022:

** **

Ch. 1 of Euler's E110: **Scientia
Navalis I : Naval Science **has now been
translated:

Ch. 2 of Euler's E110: **Scientia
Navalis I : Naval Science **has now been
translated:

Ch. 3 of Euler's E110: **Scientia Navalis I : Naval
Science **has now been translated:

Ch. 4 of Euler's E110: **Scientia Navalis I : Naval Science **has
now been translated:

Ch. 5 of
Euler's E110: **Scientia Navalis I : Naval Science **has
now been translated:

by Gregory St. Vincent

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:

** De Motu
Aeris in Tubis : Concerning the Motion of Air in Hyperbolic Conoidal
Tubes with the Minimum Disturbance of the Air. **has recently
been translated:

** **

*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. 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 * now complete. ** **link here***.**

** **

*Euler*** : E17 : The complete Ch.1- Ch.9
of Euler's arithmetic text are available for download.**

*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 **

* *

**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. * ** 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. 2**^{nd}^{ }*, ***2022 , **

*iandotbruce@acedotnetdotau** .*