*General Introduction : The
State of this Site Sept. 2018. *

**This
website is now 12 years old : There are now in excess of 880 URLs. It is pleasing to note that on a monthly
basis it attracts around 10,000 visitors, and 100,000 hits are made, and that more than 1,000 files are downloaded
on a daily basis to mathematicians and students of mathematics in around 150
countries, of which the U.S. accounts for approximately a quarter or more, 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. **

**
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 year has seen the completion of my
translation of Vol. 4 of Euler's Introduction to Integral Calculus, published
from his posthumous papers. 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 have now finished 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 *

* Occasionally people
send e-mails 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. 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. Sept. 2018.*

*Feel free to contact me for any relevant reason as discussed ; my email
address can be extracted from :*

*iandotbruce@acedotnetdotau *

**Latest addition May 10 ^{th} , 2019:**

**A new work is
started here, Euler's A
Tentative Exposition of a new Theory of Music…., E33; at present we have
the Forward and Ch. 1, concerned with the physics of generating music, Ch. 2, which initially is concerned with the
beauty of music, and later starts to investigate some aspects of its structure.
Ch. 3 is now complete, and this is essentially a brief summary of the rest of
the treatise. Ch. 4 considers concords or pleasing notes played together, later
to be called consonances. Ch. 5
continues this investigation by considering the charm of several concords, as
measured by Euler's function. Ch. 6 though short, gives some ideas regarding
the numerical equivalents of the musical structures found in a harmony, and how
to evaluate them. Ch. 7, though short
also, presents a view of the names of the various intervals in use at the time,
and some of the confusion thereof; this terminates in a table showing the
intervals in use, as well as the associated ratios with their base 2
logarithms, and a measure of the pleasantness associated with each. Ch. 8 is
concerned with a historical review of the development of the octave from ancient
Greece times, including the exponents of the various attempts, involving a
power of two with the factors 3 and 5, and with the nomenclature of the various
consonants arising. Chapter 9 is concerned mainly with an augmented version of
the diatonic-chromatic genus; this is continued in Ch. 10, though the higher orders
scales seem to be of little practical value, and to be little more than a mathematical
exercise. However, the fact that Euler was able to combine the diatonic,
chromatic and harmonic scales into a generalised scale is significant, I
think. Ch. 11 sets out the kinds of
consonances that arise in the diatonic-chromatic scale introduce, according to
the methods presented previously, starting from the simplest, detailing the
agreeability, etc. ; this is the first chapter in which Euler actually
introduced sheet music in his exposition. Ch. 12 sets out the enumerable scales
possible under Euler's classification. Ch. 13 is a detailed account of the
consonances in the diatonic-chromatic scale, including a table of the exponent
relations to the scales produced for the orders required. The final chapter
gives an overview of the usefulness of Euler's theory : **

**Another new work
is started here, Euler's Dioptrics….,
E367; now we have Chapters 1-7 translated, which is essentially the whole
of Book I; the first chapter is concerned with the spreading of the image by a
single thick convex lens, (essentially the first ever treatment of spherical
aberration), while the second considers the spreading of the image by a number
of such lenses on the same axis, with attempts to minimize this effect for two
thin lenses . Ch. 3 is rather long, but
gives a thorough discussion of how to minimize the spreading or confusion of
the final image for two, three or four thin lenses. The case for 4 lenses turns
out to be especially relevant. Ch. 4 is a compilation of the preceding
chapters, and considers the combined effects on the image viewed by the eye due
to magnification, confusion of image, and clarity. Ch. 5 is concerned with
determining the field of view of an object seen through a number of lenses, and
a convenient place for the eye. Unfortunately, the treatment of the eye's accommodation
was not understood at this time, and Euler considers the eye as a camera
obscura. In addition, the treatment of image formation is rather obscure in
this work up to this point. Ch. 6 is
long and rather involved; in it Euler sets out formulas for the distance and
height of the image formed by a series of lenses. Note that here and in the
previous chapters, some of the lines do not refer to rays, but to measurable
lengths. The effect of lenses of differing refractive indices is introduced,
and finally a method is produced for producing an achromatic final image from a
series of such lenses, as well as being free of confusion; these may be viewed
by the modern reader with some suspicion. Ch. 7 is a summary of the preceding
chapters, to which I have added the occasional notes. **

* ***Some
interesting developments were taking**** place in optics**** at this time: namely the construction of achromatic
lenses for optical instruments. John Dollon was involved in this, and objected
to Euler's theoretical handling of the problem. Euler eventually produced ***E266*** on the achromatic doublet, which I have translated here,
along with Dollond's letter to the Royal Society as an appendage (already in
English).**

**A translation of Book II of Euler's Dioptrics…., E386 has started. At present Ch.I is presented here, which is an
updated summary of Book I. Whether he understood **

**Contents.**

*Euler*** : E17 : **

*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 & E054 & E134 &
Fermat letter to Wallis, E031, E041,
E044, and E045 are present also, some of which are referred to in the
Mechanica; 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. Link to the contents document by clicking here. *

** **

*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
.**

*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 10 ^{th }, 2019, **

*iandotbruce@acedotnetdotau** .*