*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 **,

* 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: Nov. 22**^{nd} , 2022:

The work on the transcription of the conic
sections of Appolonius by Gregorius continues now with the parabola, of which
there are 8 parts : **Prolegomena ; Part 1 ; Part 2
; Part 3;
Part 4 **are presented here initially.

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:

Ch. 6 of
Euler's E110: **Scientia Navalis I : Naval Science **has
now been translated:

Ch. 7 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. *

*
Link to the contents vol.2 document by clicking
here. *

*The
translation of Euler's ALGEBRA is now complete ; Link to the contents here .*