Some Mathematical Works of the 17th & 18th Centuries, including Newton's Principia, Euler's Mechanica, Introductio in Analysin, etc., translated mainly from Latin into English

Most of the translations are from Latin by Ian Bruce, with some papers by others

See below for latest additions

General Introduction: The State of this Site June, 2022

This website is now 16 years old. There are now in excess of 900 URLs. It is pleasing to note that at the best of times on a monthly basis it attracted around 10,000 visitors, and in excess of 100,000 hits were made, and that more than 1,000 files were downloaded on a daily basis to mathematicians and students of mathematics in around 150 countries, of which the U.S. accounts for approximately half, 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. However the Covid virus has led to quiet times and the occasional very busy times, though after a drop in downloads last year, the current rate has risen far beyond the usual rate. It is most unfortunate for humanity in general that this catastrophe has happened. It has had a very bad effect on education worldwide, and of course on tertiary education.

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 few months has seen the translation of the chapters of the first volume of Euler's work Scientia Navalis, or the Science of Ships. At present Chap's 1 to 6 are available here. This follows the continued translaton of the works of Gregorius started many years ago now. At present Books 1, 2, 3 & 4 are avaialble in complete translation. The last year has seen the completion of my translation of Euler's Dioptricae: Vol. 1 on general principles, Vol. 2, on refracting and reflecting telescopes, and Vol. 3, concerning microscopes. A number of Euler's works have been completed, including fluid flow. Prior to this, a treatment of the analysis of continued fractions by Euler was given, with applications to square root extraction, etc. 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 had finished previously Lagrange's Traitè de la Resolution des Equations Numeriques de tous les Degres, in which he tackled the resolution of numerical equations of any degree. A few interesting and relevant commentaries from the time including one by Vandermonde have been introduced, from the 1770's, when attempts were being made to solve 5^th-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. I have akso translated E33, which is a tentative mathematical theory of music, produced by Euler in his early days.

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. If you feel like sending me an email for whatever reason, please do so; however you have to unscramble my email address below.

The most popular files downloaded recently not in order have been Euler's Integration, Ch. I, Newton's Axioms and the addition to Book II ch. 1 of the Principia, and some chapters of Euler's Mechanica I & II and his Introductio in Analysis, completed this last year. Parts of Huygens' Horologium are very popular, some early Euler papers, esp. E025, and various parts of Gregorius and James Gregory's Optica Promota also, 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. Occasionally there are some notes to help you along, especially at the start of the chapter.

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 avaiable, 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 10 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 translations 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 Galileo, 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, the books I & II of Euler's Introductio in analysin infinititorum, his Algebra, Methodus Inveniendi Lineas Curvas Maximi Minimive Gaudentes, dealing with variational problems of integrals, and some later related papers, E296 & E297, following Lagrange's brilliant suggestions. In addition I have added Gellebrand's second part of Briggs's Trigonometriae Britannicae. See below for later additions.

Very occasionally someone send me an e-mail, for which they have to decipher my address so constructed to avoid tedious junk mail, 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. On the other hand, if you are pleased with the translations, feel free to tell me so. 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. Jan. 2022.

Feel free to contact me for any relevant reason; my email address at present is, which you have to cut and paste:
ian.bruce@ace.net.au

Latest addition: May 17, 2023:

Some new Euler translations: E27; E28.

The work on the transcription of the conic sections of Appolonius by Gregorius continues now with the parabola, of which there is an introduction and the 8 parts are presented here: Prolegomena; Part 1; Part 2; Part 3; Part 4; Part 5; Part 6; Part 7; Part 8.

Ch. 1 of Euler's E110: Scientia Navalis I: Naval Science, has now been translated: Naval Science I CH. 1.
Here Euler considers the static cases of prisms of various cross-section floating on water. This leads to the number of ways a given prism can float on water, either vertically or longitudinally.

Ch. 2 of Euler's E110: Scientia Navalis I: Naval Science, has now been translated: Naval Science I Ch. 2,
Here Euler for most of the chapter provides a summary of the principles of dynamics needed to understand the following chapters. He is interested especially in considering the weight of the vessel acting downwards through the centre of gravity, and the upthrust generated by the weight of water displaced, which acts at another point, which is variable, and rotations about 3 principle axes are considered.

Ch. 3 of Euler's E110: Scientia Navalis I: Naval Science, has now been translated: Naval Science I Ch. 3.
This is a long chapter running to around 80 pages, in which initially Euler establishes the principles by which floating bodies may be disturbed from their resting position; various prisms with different cross sections are discussed; later the discussion turns to ship like curves where again the principles of equilibrium are discussed. This is a challenging chapter, and shows Euler's amazing mathematical powers.

Ch. 4 of Euler's E110: Scientia Navalis I: Naval Science, has now been translated: Naval Science I Ch. 4.
This chapter outlines five of the main features of a ship that need to be examined carefully. These include the depth of the centre of gravity of the ship, the effects if being rotated about the three principle axes, and the effect of the rudder.

Ch. 5 of Euler's E110: Scientia Navalis I: Naval Science, has now been translated: Naval Science I Ch. 5.
This chapter provides a great deal of information in anticipation of Ch. 6. The mathematical means according to Euler are established for calculating the resistance of a number of elementary shapes moving through water. This includes numerous side issues where particular integrals are evaluated. This is a long chapter of 78 pages!

Ch. 6 of Euler's E110: Scientia Navalis I: Naval Science has now been translated: Naval Science I Ch. 6.
The mathematical means according to Euler are established for calculating the resistance of a number of elementary shapes moving through water, making use of Newton's rule for the resistance of an object moving straight through a fluid such as water at rest. Different kinds of geometrical shapes are taken to represent the bows of the ship, where three curves are taken to delineate the shape. The resistance is found by resolving double integrals over the surface ofor a given shape, e.g., part of a sphere, cone, triangle, etc. This is another long chapter of 90 odd pages, which has taken some 4 months to translate. Chapter 7 does not seem to add much new, and will be handled more slowly with other projects in hand.

Ch. 7 of Euler's E110: Scientia Navalis I: Naval Science has now been translated: Naval Science I Ch. 7,
This is the final chapter of Euler's first extended work on ships, It is concerned with the motion of bodies floating or ships sailing in water. Numerous formulas are developed for expressing the forces acting on ships due to water resistance, wind on sails, use of oars, etc., and the paths taken are calculated; there is a great deal worked out in this chapter from basic principles, according to the state of physical knowledge at the time.

QUADRATURAE CIRCULI
by Gregory St. Vincent

This is the start of a truly mammoth book running to some 1250 pages. At present only Books I, II, & III have been translated here. The work received a lukewarm reception at the time (1647) as Gregorius asserted that he could square the circle, as the title indicates. However, there is a place for this work in the history of mathematics, as it was one of the forerunners of the theory of integration, and the natural logarithm was developed from geometic progressions applied to hyperbolic segments - though the present work does not extend this far.
The introduction to Conic Sections, the Prolegomena, is now presented here in addition to Books 1, 2, & 3. Gregorius has made a slightly different classification of cones than that of the Ancients, which is presented here and compared with those. In addition, Book 3 of Gregorius' work Quadrature of the Circle is now complete. This book is concerned with the further development of classical Greek geometry applied to circles. It is some time since Books 1 and 2 appeared, and the method of presentation has evolved a little since then, so that the current method of presentation has been adopted.

Book1: Proportions between line segments Book2: Geometrical Progressions

Book3: Circles Prolegomena Book4: Ellipses, Parts 1 to 6

Ch. 5 of Euler's De Motu Aeris in Tubis: Concerning the Motion of Air in Hyperbolic Conoidal Tubes with the Minimum Disturbance of the Air has recently translated: etr424ch5.pdf.
Ch. 5 is concerned with finding the equations for the steady-state flow of air in tubes of this nature gradually increasing in cross-sections. Generally Euler finds that such tubes do not make good as musical instruments, unless the initial speed of the air is zero, though trumpets, bugles, and horns do find a place in reality.

Ch. 6 of Euler's De Motu Aeris in Tubis: Concerning the Minimal Motion of Air in Conoidal Tubes has now been translated, etr424ch6.pdf.
This is the final chapter in this series, and involves the comparison of conoidal and cylindrical tubes.

Euler spent some time showing how to produce theorems relating to the expansion of trigonometric functions of some multiple of an angle raised to some power as series involving simple sines and cosines of angles, such as in e246.pdf presented here. In addition, we now have e061.pdf, in which a new method is found for expanding the product of π with the sines and cosines of any angles as infinte series of the powers of the reciprocals of whole numbers.

Recently translated Euler works such as the Optics can be accessed below.

CONTENTS

Lagrange's Traité de la Resolution des Équations Numériques de tous les Degré is now complete, including Notes I-XIV; E30, E282, and Vandermonde's Resolution of Equations.
Link here

Hermann's Phoronomia (1715).
Link to the contents document by clicking 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.

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!

Some Mathematical Works of the 17th & 18th Centuries, including Newton's Principia, Euler's Mechanica, Introductio in Analysin, etc., translated mainly from Latin into English

Most of the translations are from Latin by Ian Bruce, with some papers by others

See below for latest additions

QUADRATURAE CIRCULI by Gregory St. Vincent

QUADRATURAE CIRCULI
by Gregory St. Vincent