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, Huygens' Horologium Part I, Vieta's Sectiones Angulares, some early Euler papers, and various parts of Gregorius, and there is of course a constant demand for material on logarithms. 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 went to school there in the late 1950's and early '60's), with a background in maths and physics (I have a number of publications in various journals, such as Math. Gaz., Fibonacci Q., Am. J. Phys., Eur. J. Phys, etc.) : 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 am translating Euler's Mechanica at present, gradually. 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.]

    Happy browsing! July '08. I. B.

    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. E031, E041, E044, and E045 are now present also, as they are referred to in the Mechanica 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.
  • A translation of E016, Book 2 of Euler's Mechanica has commenced. At present the preface, ch.1 and ch.2 are available; the latter is in sections a - g, as it is a very long chapter. Ch. 3a, 3b, 3c, 3d & 3e are now completed. 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.
  • 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.