*With apologies to Samuel Johnson, on regarding his
dictionary, with some small changes : **It is the fate of those who toil at the lower
employments of life, to be rather driven by the fear of evil, than attracted by
the prospect of good ; to be exposed to censure, without hope of praise; to be
disgraced by miscarriage, or punished by neglect, where success would have been
without applause, and diligence without reward. Among these unhappy mortals is
the translator of Latin mathematical works of days gone by; whom mankind have
considered, not as the pupil, but the
slave of science, the pioneer of mathematics, doomed only to remove rubbish and
clear obstructions from the paths through which Learning and Genius press
forward to conquest and glory, without a smile on the humble drudge that
facilitates their progress. Every other author may aspire to praise; the
translator can only hope to escape reproach, and even this negative recompense
has been granted to a very few....*

*General Introduction : The
State of this Site Sept. 2017: Annual Report. *

**This
website is now 11 years old : There are now in excess of 800 files available
for download. 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 : I am
honoured to be able to perpetuate his memory here, and the chapters of Euler's
E842, except for one chapter, missing
from the original text, which he translated from Euler's German, at the age of
93, are downloaded on a regular basis.**

** ***Hermann's
Phoronomia has now been moved into the general
scheme of things. The last year has seen the*

*completion 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 Équations 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. 2017.*

**Latest addition July 5 ^{th} , 2018:**

**A new work now
begins, Euler's E17, Arithmetic, or
the General Art of Reckoning; at
present the forward and chapters 1, 2, 3 and 4 have been translated from
Euler's German. These early chapters give instructions on how to write and say
numbers, addition, subtraction, and
multiplication. It is interesting to see how Euler tackles what would appear to
be a task thrust upon him and to be way below his abilities….. However, I find the
way in which Euler thought about numbers to be very interesting, as is his
instruction technique: Most of us learn how to do arithmetic at an early age without
necessarily understanding it completely. See immediately below for the link.**

**The translation of Lagrange's Treatise on the Resolution of Numerical
Equations of all Degrees is now**

** complete,
and is presented below . E30
& E282 are relevant paper produced by Euler ; The latest addition is a
translation of A.T. Vandermonde's On the resolution of equations ; this is a longer paper, originally presented
in 1771 to the French Royal Academy, in which a general scheme was presented
for resolving equations of any order, but which failed on the fifth order and
thereafter, these three papers
are now included below in the Lagrange link.**

**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.__**
July 5 ^{th }, 2018, **