** With apologies to Samuel Johnson: **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. 2012. *

**However, not withstanding the similarities of the
present task with 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. The most popular files downloaded recently
not in order have been Euler's Integration Ch. I , and some chapters of the
Mechanica I & II , parts of Huygens' Horologium are very popular, some early
Euler papers, esp. E025, and various parts of Gregorius and James Gregory's
Optica Promota, and there is of course a constant demand for material on
logarithms, and Harriot's book salvaged from his posthumous notes is of
interest to browsers. Lately Jim Hanson's work on Napier's Bones and Promptuary
have been very popular. This site is unique in that it provides the only
translation available into English of a number of important works. Texts are
presented with the understanding that they cannot be error-free, and reveal the
translator's idiosyncrasies to some extent. Usually there are some notes to
help you along, especially at the start. **

*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. Work on Newton's Principia
is now completed; 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 Newton's methods are obscure, and it has pleased me to be able
to unravel some of these. The Principia never was, and never will be, an 'easy
read'; however, Newton was a man from the new age of mathematical science
initiated by Kepler and Galilio, opening a hidden door via calculus, and who
ushered in almost single handedly the world of the mathematical analysis of
physical phenomena: Newton's view of the world, controversial at the time
because of the idea of action at a distance,
is still the one that we accept largely. To our store of translations
also has been added Napier's De Arte Logistica, and the first book of Euler's
Introductio in analysin infinititorum, while work is almost finished on the appendices
to the second book.*

* Occasionally people send
e-mails concerning things they are not happy about in the text, and their
suggestions may be put in place, if 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 6
years old! *

*Happy browsing! IAN BRUCE. August 2012.*

*Latest
addition May 3rd, 2013: I have started on
a translation of one of Euler's works E65 on the max. & min. curves initially
relating to various simple conditions, and extending to calculus of variations
type problems, of which this work is a forerunner ; at present Ch. 1, Ch. 2 & Ch. 3 of Methodus
Inveniendi Lineas Curvas Maximi Minimive Gaudentes……… have been
prepared; see below; previous to this, I have been busy with Euler's papers
(E305) , (E306), (E307) on the propagation of sound ; see below. Previous to this I have put in place a translation of
Euler's paper E524 relating to spherical triangles, which in a very short and
straight-forwards manner derives all the formulas relating to spherical
triangles, and is well worth reading if your knowledge is hazy or lacking in
such matters ; in addition, the second part of the Trigonometria Britannica by
Henry Gellibrand has now been translated and presented here; concerning the
solution of plane and spherical triangles by using logarithms, essentially by
using Napier's circular parts. In addition, a new online version of
Euler's classic work Introductio in Analysin
Infinitorum is presented here : see below. This completes the main text
books of Euler on Calculus and Introductory Analysis. Prior to doing this, I
had finished translating a long
forgotten work of John Napier : De Arte Logistica……. *

*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. *

*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, & 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
underway. At present you can access Ch. 1, Ch.2a & 2b, Ch. 3 by clicking: Link toMaxMin.*

*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. *

*An annotated
translation of Christian Huygens' Pendulum Clock
is presented. 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.__* **May 3rd , 2013, **latest revision. Copyright : I reserve the right to
publish any translated work presented here in book form. However, if you are a student,
teacher, or just someone with an interest, you can copy part or all of a work
for legitimate personal or educational uses. Please feel free to contact me if
you wish by clicking on my name here Ian Bruce.,
especially if you have any relevant comments or concerns.*