** 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 Oct. 23rd, 2014: A new series of translations is now underway
involving Leibniz’s calculus; initially we will concentrate on his introduction
of differential calculus to the mathematicians of the time in the *Acta** Eruditorum* (*AE*) : The time sequence of the papers is
used to reference these; see below for the papers, the first translated is the
Nova Methodus *AE*13,
followed by Optics, Catroptrics, and Dioptrics from a single principle *AE*9, Concerning the rectification of areas *AE*11; *AE*19 Concerning a Recondite Geometry and the analysis of the
indivisible and the infinite;
*AE*13 on constructing the integrals of curves mechanically, in which
also he introduces the *Tractrix** * curve; *AE*6, deals with the calculation of π
from infinite series; *AE*4, concerning osculating circles; AE7,
Optical curved etc.

Previous to this,
I have now 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.

*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 AE1, AE4, AE6, AE7, AE9, AE11, AE13 & 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.__* Oct.23rd ^{ }, 2014, *