** 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. 2014: Annual Report. *

**However, notwithstanding the similarities of the
present task with Dr. Johnson's remarks about compiling his dictionary, it is
pleasing to note that for this website, around 3500 visits and 50,000 hits are
made on a monthly basis, and that around 25,000 files are downloaded monthly 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. Not
much has changed over the past year; I had finished translating Euler’s ***Neue** Gründsatze der Artellerie **some time
ago, and now at last I can draw a deep breath again after attending to Daniel
Bernoulli’s **Hydrodynamicae**, which
has now been completed. 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.*

** 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 18 months 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. The site
is now 8 years old! *

*Happy browsing! IAN
BRUCE. Sept. 2014.*

Latest addition Sept. 26^{th},
2014: A new series of
translations is now underway involving Leibniz; initially we will concentrate
on his introduction of differential calculus to the mathematicians of the time
in the Acta Eruditorum : see below for the first
paper, the Nova Methodus (*AE*13), as well as the second Optics, Catroptrics, and Dioptrics from a single principle (*AE* 9), a third, Concerning
the rectification of areas (*AE*11);
and no.4: (*AE*19) Concerning a Recondite Geometry and
the analysis of the indivisible and the infinite, which is one of Leibniz's most famous papers, as he finally
gets round to providing some information about integration….AE refers to
the article number in the journal *Acta Eruditorum*.

Previous to this, I have now completed Ch. XIII of Daniel Bernoulli's Hydrodynamicae here in translation, this last chapter considers the action: reaction nature of water flowing out of a cylinder; if you are looking for what was to become the Bernoulli Principle in fluid dynamics, then to some extent you will be disappointed; you have to thank the quirky humor of Euler for this, although Bernoulli’s work laid the foundations for that of Euler; you can find the link in the Bernoulli section below. This work sets the foundations of the science of the same name, in which Daniel Bernoulli combines his remarkable mathematical skills with experiments to put a difficult subject on a firm foundation; most of the work was done at St. Petersburg and viewed by Euler with interest, when eventually published. Prior to this, a translation of Euler's E248 is now presented here accessible from the Euler papers link below: this is an attempt by Euler to provide a theory for that ancient device for raising water: the Archimedes Screw, which Daniel Bernoulli provides in Ch. 9 of his work, and which I thought might be of interest – the difference in the methods of tackling this machine; the one a mathematician and the other a physicist. Prior to this I have completed a translation Euler's Neue Gründsatze der Artellerie we have inserted a small paper by Euler from his Opera Postuma along with the main translation, (E853) dating from the early days in St. Petersburg, where he witnessed the vertical firing of a cannon conducted by Daniel Bernoulli, given serious attention in his later work, which is of some interest.

*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 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 practise 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 labourous
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 AE13, AE9, AE11, & 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, 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
is now complete. You can access these by clicking: Link to volume
I or Link to volume
II , or Link to volume
III.*

*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. *

*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.__* Sept. 19 ^{th} , 2014, *