** 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. I have moved on to translating Leibniz’s articles on
his calculus presented in the Acta Eruditorum from 1682 onwards. *

** ***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 July 22th, 2015:** A series of
translations is underway on the ** Phoronomia** of Hermann, a work that
bridges the gap between Newton and Euler; at present Book I and Book II,
Section I, Ch's. 1 - 7, the former dealing with statics and dynamics, the latter
the beginnings of hydrostatics, are presented here, with some introductory
material. The latest chapter is concerned with the proportionality between the pressure
and density of air. You can find the link below near the top. Previous to this,
a series of translations has been undertaking involving Leibniz’s calculus, the
latest being

Previous to this,
I had completed translating the final chapter Ch. XIII of Daniel Bernoulli's *Hydrodynamicae*, which 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. This summarizes the translations over the last
two years or so.

*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, 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.__* July 22th ^{ }, 2015, *