Some Mathematical Works of the 17th &18th Centuries :

    Translated from Latin by Ian Bruce.

    Introduction.

    I am pleased to tell you that in the last 12 month period some 125,000 hits have been recorded, and over 17,500 visits have been made to this website. The website is thus on a par with small select sites, and has around 50 - 60 visits daily. The most popular files downloaded not in order have been Euler's De Sono, and some chapters of the Mechanica I & II , Huygens' Horologium Part I, Vieta's Sectiones Angulares, the paper on Lunes, 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. This site is unique in that it provides the only translation of a number of important works. Although I have endeavoured to the best of my ability to render the original Latin texts into their equivalent in modern English, usually in the present tense as in modern mathematical usage with disregard for the subjunctive, future and past tenses so beloved in latin, the texts are presented with the understanding that they cannot be error-free, and reveal my own idiosyncracies to some extent. Usually I provide some notes to help you along, especially at the start.

    The site is produced, funded, and managed by myself, a retired person living in Adelaide, S. Australia, ( I originated in the N.E. of Scotland near the town of Ellon, and grew up there in the late 1950's and early '60's; if you are an old mate, send me an e-mail!) ; I have a background in maths and physics (I have a Ph.D in atomic physics from Flinders University, and a number of publications in various journals, such as Math. Gaz., Fibonacci Q., Am. J. Phys., Eur. J. Phys, etc. over the last 30 years) : my 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 : I have recently finished translating Euler's Mechanica , and I am now working on his Tractus de Motu Corporum Rigidorum..... Occasionally people write to me concerning things they are not happy about in the text, and their acknowledged suggestions are put in place. If you feel that there is something wrong somewhere, please let me know. Finally, if you are a person involved in using the contents of this site in some way as class exercises, then I would like to hear from you, just so that I know what use is being made of the material contained here, and note any comments you might wish to make. [See the links on the contents pages.]

    We are now two years old! Happy browsing! Jan. '09. I. B.

    June 24th, 2009 latest revision.

    Latest additions starting from the most recent : The translations continue : Tractus de Motu Corporum Rigidorum Ch's. 1 - 17 of Euler's treatise on the motion of rigid bodies; Ch. 17 applies Euler's equations to the theory of tops. A translation of the O. O. Mechanica editor's forward has been added by E. Hirsch. Jim Hanson's work on Napier's Promptuary and Bones is in place, with a few other items in the Napier index. You need broadband to download the picture files quickly, of which there are many.

    Contents.

  • 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 some interesting identities due to Napier. Link to the contents document by clicking here.
  • 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.
  • 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. No latin text provided. 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.
  • Problems relating to isochronous and brachistochrone curves are presented in E001 and E003; a dissertation on sound in E002; 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 cantenaries 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 differerential 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, E031, E041, E044, and E045 are present also, some of which are referred to in the Mechanica. Also papers by Lexell tr. by J. Sten appear here.Link to the contents document by clicking here.
  • My translation of E015, Book I of Euler's Mechanica has now 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.
  • A lot of the work has been done in translating Euler's next major contribution to mechanics (E289): Theoria Motus Corporum Solidorum seu Rigida. At present Ch's 1 - 6 concerning the general theory of the motion of point masses is in place , and Ch's. 1 - 10, comprising all of Vol. III of Series II of the Opera Omnia. and Ch's. 11 - 17 of Vol. IV Link to the contents document by clicking here.
  • I am pleased to place on the Euler index page a recent translation of E698 (on the angular excess of spherical triangles) by Johan Sten. In the same place you can also find the paper by Lexell that inspired Euler's paper, also presented by Johan Sten. Johan has been busy again, and has translated the paper dealing with Euler's Angles, E478; and another paper by Lexell on the motion of a rigid body.
  • 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.
  • An annotated translation of Section VIII of Book II of Newton's Principia on sound is presented. 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 trigonometrical 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. June 14th, 2009 latest revision. Copyright : I reserve the right to publish any translated work in book form. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for legitimate personal or educational uses. Please feel free to contact me if you wish by clicking on my name, especially if you have any relevant comments or concerns.