*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: May 17**^{th} , 2023:

Some
new Euler translations : **E27** ; **E28**

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:

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 p with 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,
E23, E24, 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* .* *

** ***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 **

* *

**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. *