General Introduction : The
State of this Site Aug. 2019.
This
website is now 13 years old : There are now in excess of 880 URLs. It is pleasing to note that on a monthly
basis it attracts around 10,000 visitors, and in excess of 100,000 hits are
made, and that more than 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. This amounts roughly to a 500 page book being
printed from the website worldwide every
15 minutes. 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 after
arriving in Australia on the Dunera during WWII : I am honoured to be able to
perpetuate his memory here in several works which he translated from Euler's
German, at the age of 93.
The
last year has seen the completion of my
translation 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.
I have now finished Lagrange's Traité de la Resolution des Équationes Numériques de
tous les Degrés, in which he tackles the resolution of numerical equations of any
degree. A few interesting and relevant commentaries from the time including one
by Vandermonde have been, from the 1770's, when attempts were being made to
solve 5th order equations. In addition I have translated E17, which
is really an introduction to the theory behind arithmetical calculations; the
applications are presented a little later, relating to business calculations ,
exchange rates and so on. This I have not translated so far. I have recently
finished E33, which is a tentative mathematical theory of music, produced by Euler in his early days; since
then I have been involved rather deeply with the second volume of Euler's
Optics, which I have just finished translating. Unfortunately, Euler had chosen to ignore the physics largely and
turned his version of lens and telescope design into a mathematical exercise
where the components were not tested individually as the analysis progressed,
and which finally does not appear to have produced telescopes that worked
particularly well; he blamed the artificier for the deficiencies.
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 link below. 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. Aug. 2019.
Feel free to contact me for any relevant reason as discussed ; my email
address can be extracted from :
iandotbruce@acedotnetdotau
Latest addition Sept. 16th , 2019:
A new work has
been completed here in part,
Euler's Dioptrics…., E367; now we have Chapters 1-7 translated, which is
essentially the whole of Book I; the first chapter is concerned with the
spreading of the image by a single thick convex lens, (essentially the first
ever treatment of spherical aberration), while the second considers the
spreading of the image by a number of such lenses on the same axis, with
attempts to minimize this effect for two thin lenses . Ch. 3 is rather long, but gives a thorough
discussion of how to minimize the spreading or confusion of the final image for
two, three or four thin lenses. The case for 4 lenses turns out to be
especially relevant. Ch. 4 is a compilation of the preceding chapters, and
considers the combined effects on the image viewed by the eye due to
magnification, confusion of image, and clarity. Ch. 5 is concerned with
determining the field of view of an object seen through a number of lenses, and
a convenient place for the eye. Unfortunately, the treatment of the eye's
accommodation was not understood at this time, and Euler considers the eye as a
camera obscura. In addition, the treatment of image formation is rather obscure
in this work up to this point. Ch. 6 is
long and rather involved; in it Euler sets out formulas for the distance and
height of the image formed by a series of lenses. Note that here and in the
previous chapters, some of the lines do not refer to rays, but to measurable
lengths. The effect of lenses of differing refractive indices is introduced,
and finally a method is produced for producing an achromatic final image from a
series of such lenses, as well as being free of confusion; these may be viewed
by the modern reader with some suspicion. Ch. 7 is a summary of the preceding
chapters, to which I have added the occasional notes. Ch.1-Ch.7 are now complete.
Some
interesting developments were taking place in optics at this time: namely the construction of achromatic
lenses for optical instruments. John Dollond was involved in this, and objected
to Euler's theoretical handling of the problem. Euler eventually produced E266 on the achromatic doublet, which I have translated here
following the link below, along with Dollond's letter to the Royal Society as
an appendage (already in English).
A translation of Book II, Sect. 1 of Euler's Dioptrics…., E386 is now complete. Ch.I
is an updated summary of Book I. Whether he understood Dollond's new approach for making achromatic lens
doublets is not clear; though some clarification has been made of other
matters. Ch. 2 more or less gives Euler's
recipies for making perfect composite objective lenses for telescopes; i.e. those with the minimum confusion or
spherical aberration and free from colored fringes or achromatic; two or three
lenses of differing refractive indices are used. Ch. 3 gives an attempt to
classify types of telescopes according to the number of real intermediate
images produced. Ch. 4 is concerned with perhaps the simplest of telescopes,
namely a convex objective followed by a concave eyepiece ; this kind of
telescope is treated at length, although it would appear to be at variance with
the actual devices produced by Dollond; yet there appears to be much of value
in this text, though Euler seldom gets a mention in optics text re telescopes.
Ch. 5 is now complete, which shows how to perfect simple upright image
telescopes by the use of extra lenses; this is rather a long chapter.
Book II, Sect. 2, Ch. 1 details the construction of telescopes
that may have a compound objective as well as a possible compound eyepiece
lens. I am including some extra background material at this stage; Euler's Dioptrics is difficult to follow in parts, due to the algebraic
methods used, which are not always correct, as indicated in King's book The History of the Telescope; that does
not mean, of course, that the work should be neglected, as has been the case
until now.
Book II, Sect. 2, Ch. 2
considers the advantages of inserting an extra lens to coincide with the
primary image formed by the objective lens, and the nature of this lens to
reduce confusion (i.e. aberrations);
glass of the same refractive index is considered throughout at this stage. Ch.
3 considers further improvements using two kinds of glass with differing
refracting indices, forms of crystal and crown glass. A formula for the
dispersion of each kind is introduced and applied to the individual lenses; as
mentioned above, defects in the final design are attributed to the poor
artificier employed to implement the designs, which did not appear to work very
well, and did not produce the sharp colour free images intended. The
translation of Part 3 of Book 2 is now underway, and at present we can now
present Ch.1 of this part, which is concerned with producing upright images, by
the addition of extra lenses. I have included some notes here; Euler's work was
continually frustrated by the imperfect telescopes his artisans could make from
his theoretical calculations; the flaw would appear to arise from his use of a
formula for the focal length of the objective which he produced in Ch. I of the
first volume, long before the different forms of lens aberration were
understood.
Contents.
Euler: E33 : A Tentative Exposition of a new Theory of
Music…., the whole work, Ch.1-Ch.14,
is now complete. link here.
Euler : E17 : The complete Ch.1- Ch.9 of Euler's arithmetic text are available for download. link here
Lagrange
Work: 'Traité
de la Resolution des Équations Numériques de tous les Degrés' is available now complete. Including
Notes I-XIV; E30, E282, and Vandermonde's Resolution of Equations are presented: 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 ; E675
& E640 ; E595; E391; E581; Supp. 8a
& 8b, E506 & E676; Supp. 9 & 10, E677, E678, E679, E680,
and E681, while Supp. 11 consisting of E420,
are presented in
Vol. IV, attached to the end of Vol. III. You can access these by clicking: Link to volume I or Link to volume II , or Link to volumes III & IV
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.
An 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. Sept.
16th , 2019, 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
private personal or educational uses. Any other form of distribution is illegal
and has not been permitted by the author, Ian Bruce, who asserts that the
contents of this website are his intellectual property: You are NOT given the right to sell items from this website
on the web or otherwise offer 'free' to download in any shape or form, such as
e-books. Be very wary of websites that make such offers, there are a
few around, usually trying to make money out of someone else's work! This
website presents a genuine attempt to provide much mathematics lost to the
modern reader, who is unskilled in Latin. Please
acknowledge me or this website if you intend to refer to any part of these
translations in a journal publication or in a book. Feel free to
contact me for any relevant reason as discussed above ; my email address can be
extracted from :
iandotbruce@acedotnetdotau .