*General Introduction : The
State of this Site Oct. 2021. *

**This
website is now 15 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
Corona 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 **

**
*** 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 are here presented in complete translation, while a start
has been made on Book 4 on Conic Sections: Ch's 1-5 on the ellipse are
presented here.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. Nov. 2020.*

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

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

* I can be found also at: ianb05965@gmail.com *

**Latest addition
Oct. 17 ^{th} , 2021: **

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** ;*

* ** ellipse part 1** ; ellipse part 2*** ;Ellipse
part 2 considers sectors and segments of ellipses; **:

* ellipse part 3; Ellipse part 3 considers
the similar and dissimilar properties of the conjugate axes and diameters of
the ellipse *

* ellipse part 4 ; Ellipse part 4
considers the basic theorems related to the focal points, the reflection of a
ray through on focus by a tangent to the other focus, and so forth;
interesting, as old-fashioned geometry alone was use; analytical methods were
just on the mathematical horizon at this time*

* ellipse part 5**: Ellipse part 5 considers various ways of
generating an ellipse. *

** 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. *

*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 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 **

* *

**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.*** Oct. 17 ^{th }, 2021, **

*iandotbruce@acedotnetdotau** .*