** With apologies to Samuel Johnson, and not to be taken too seriously : **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 literature, 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. 2016: Annual Report. *

**This
website is now 10 years old : I can only say, How time flies when you're having
fun! It is pleasing to note that it attracts around 3500 visits, and 50,000
hits are made, on a monthly basis, and that around 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. There is, of course, some seasonal variation depending on
semester demand. **

** One of my former colleagues at Adelaide
University, Ernst Hirch, 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, 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 Also, Euler's Algebra is now on this site
; Euler can make even the most elementary of mathematics interesting; which should be useful for those who use the
site as a teaching aid. The present translations involve the beginnings of
Lagrange's Works, and also of Vol. 4 of Euler's Introduction to Integral
Calculus, published from his posthumous papers.*

** ***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 3 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 hyperlink. 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. 2016.*

**Latest addition April. 20 ^{th}, 2017:** Supplement 10, comprising (E680) & (E681), is now
complete ; the first supplement is concerned with the integration of higher
order differential equations between two variables with continued differentials
present according to certain simple rules, while the second is devoted to a
means of producing a series involving binomial coefficients extending to
fractional orders starting from a special kind of differential equation of some
high order. Previously all the parts of
Supplements 1 to 9 of

**Contents.**

*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 complete. Supplements 1, 2, 3a, 3b, 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, 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
.*

*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.__* ***April 20 ^{th }, 2017, **