** With apologies to Samuel Johnson, on regarding his
dictionary, with some small changes : **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 mathematics,
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. 2017: Annual Report. *

**This
website is now 11 years old : There are now in excess of 800 files available
for download. It is pleasing to note
that it attracts around 10,000 visitors, and 100,000 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, Ernest Hirsch,
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. The last year has seen the*

*completion of Vol.
4 of Euler's Introduction to Integral Calculus, published from his posthumous
papers. I have now completed Euler's Opuscula
Analytica, the last text Euler completed while
alive, and in which he wished to draw attention to certain matters he
considered noteworthy. Now I have moved on to Lagrange's *** Traité de la Resoluton
des Équations Numériques de tous les Degrés**,

** ***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 5 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. However, I am not disposed to indulge in
idle chitchat with the serial pests who annoy me occasionally . 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. 2017.*

**Latest addition Jan 17 ^{th} , 2017:**

A
new series of translations of the works of Lagrange is now underway; I have
previously presented a few chapters from Book I, which are still available;
however, on further consideration, it seems more useful initially to translate
texts appearing in the later Books of the collected works, and in particular
initially a translation of his *Treatise
on the Resolution of Numerical Equations of all Degrees. *The reason for
this is simply that, like Euler, Lagrange kept working on earlier ideas
presented in earlier papers, and in the last few years of his life, he set out
to simplify some of his work, to make it more accessible. However, even this
has not been done in a straight forwards manner, as the detailed explanations
are assigned to Notes in the latter part of the book, following statements of
the main theorems, and examples. Lagrange had come to the conclusion, that
generally numerical methods were required to solve numerical equations, and
that general solutions in terms of coefficients, even if they were found for
degrees greater than 4, would be extremely tedious to evaluate numerically…..
At present Ch. I-VI, Ch. VI on the use
of continued fractions, are presented together with two Introductions, one by
Poinsot which gives an account of the state of algebra around the year 1800,
and the other by Lagrange himself. A few minor changes have been made to the
first 4 chapters, clarifying paragraph numbers and note numbers within
paragraphs, which are rather vague in the original text. We should note also
that some minor changes are made from the original publication and that in the
collected works.

**Contents.**

*Lagrange
Works in progress:** Ch.I – V, Ch. VI articles 1-2, 3-4 of '**Traité de la Resoluton des
Équations Numériques de tous les Degrés'** is available: link here*

*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, & IV
complete. Supplements 1, 2, comprising E670, 3a is E421, 3b is** E463**, 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, i.e. the Foundations of the Calculus of Variations, 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
.*

*The
translation of Euler's Opuscula
Analytica Vol. I is now complete***;*** being **E550
**to E562 inclusive, together with E19
and E122 *;*the sections of Vol. II E586,
E587, E588&9, E590, E591, E783, E592, E595 ***[ E594 is already present
as Supp. 5e in Vol. IV of the Integral calculus] , E596, E597, &E598, E599, & E600 are also
presented in the same contents folder as a direct follow-on. 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.__**
Jan.17 ^{th }, 2018, **