Come readers of the Cynic fect do not fay that his pamphlet is a perpetual eulogy, by implication, on the author's own dexterity in collecting and arranging anecdotes. DB-y. ART. IV. Scriptores Logarithmici; or, a Collection of several curious Tracts on the Nature and Conftruction of Logarithms, mentioned in Dr. Hutton's Hiftorical Introduction to his new Edition of Sherwin's Mathematical Tables: Together with fome Tracts on the Binomial Theorem, and other Subjects connected with the Doctrine of Logarithms. 4to. 2 Vols. 2l. 12s. od. Boards. White. 1791. MANY valuable treatifes in various parts of science must be inevitably loft, or would be altogether ufeleis, if they were not rescued from oblivion by methods fimilar to that which has been adopted by Baron Maleres, the worthy editor of the prefent collection. Some treatifes of this kind are probably dif perfed in voluminous publications, or preferved only in the archives of public libraries; fo that they are difficult of accefs, and few are able to avail themfelves of the benefit of perufing them. One or other of thefe circumftances is likely to occur with refpect to mathematical tracts of an abftrufe kind, of which the number of copies originally printed may have been small, and the fale of which will feldom defray the charge of republication. A judicious felection and arrangement of those, which are the most important and useful, muft therefore be a very meritorious undertaking, but cannot be executed without very confiderable expence and labour. The liberality of the editor is no lefs confpicuously entitled to acknowlegement on the present occafion than his love of fcience, as well as the time and pains which he must have devoted to a work of this nature. The fubject, to which his attention has been particularly directed, ist exprefled in the title, as above. The invention of logarithms, it is well known, forms a very diftinguishing æra in the history of mathematical fcience. How much they expedite every kind of calculation; to what an aftonishing degree they facilitate the explication of trigonometry to every fcience that is connected. with it; and to what an extent they have contributed to enlarge the sphere of our refearches and knowlege in this way; it is altogether needlefs to mention. Those who have any acquaintance with this branch of mathematics will not hefitate to allow its peculiar utility and importance; and they will trace with pleasure the various fteps that have been purfued by mathematicians, of all countries, in the cultivation and improvement of it. The learned editor was induced to form this collection by the perufal of Dr. Hutton's very curious hiftorical introduction to to his edition of Sherwin's Mathematical Tables, publifhed in 1785. This introduction, which contains a variety of interefting information on the fubject of trigonometry and logarithms, is reprinted at the beginning of this collection. As most of the tracts, (fays the editor,) that have been written on the latter of these fubjects, namely, the doctrine of logarithms, fince the publication of Briggs's Arithmetica Logarithmica and Trigonometria Britannica, and that are mentioned in Dr. Hutton's introduction as deferving notice, are but fhort, I thought it would be by no means impracticable to colle&t them all together into one book; and I did not doubt that fuch a collection of them, ranged in the fame order in which they were originally published, and in which, for the most part, they had been mentioned in the aforefaid hiftorical introduction, would prove an acceptable prefent to all the lovers of thefe fciences. With this view I undertook the prefent publication, in which I have omitted the two tracts of Briggs above mentioned, namely, the Arithmetica Logarithmica, and the Trigonometria Britannica, on account of their length (they being fmall folio volumes), and the tract of Baron Napier, called Canon Mirificus Logarithmorum (though it is but a fhort one), becaufe it was published before the faid treatifes of Briggs. But thefe omiffions are pretty well fupplied by the ample accounts which Dr. Hutton has given of the contents of thefe valuable pieces in his hiftorical introduction, which is here reprinted. The prefent collection begins with the two tracts published on this fubject by the famous John Kepler in the years 1624 and 1625, in which (as Dr. Hutton obferves) that very elegant and accurate geometrician has treated of logarithms according to the true and genuine idea of them, as being measures of ratios, or proportions, and has delivered his whole doctrine concerning them in a very full and scientifick manner. - < When this collection of tracts was first undertaken, I had thought that they might all have been comprised in one volume, quarto. But, as fome of the tracts were written in a very obfcure ftyle and manner, and feemed much to ftand in need of explanation; and, as they were alfo founded on a fuppofition of the truth of the famous Binomial Theorem, both in the cafe of integral powers and in the cafe of roots, which theorem but few mathematical authors have attempted to demonftrate;-I refolved to endeavour to fupply thefe defects as well as I was able, partly by adopting and inferting a demonftration of the faid binomial theorem, both in the cafe of integral powers and in the cafe of roots, that had been published by the late very learned Mr. John Landen of Walton, near Peterborough in Northamptonshire, in the years 1758 and 1764, and by adopting fome other hints given by other authors on the fame fubject, and partly by notes and tractś written by myself to answer the fame purpose, and on which (as the reader will eafily perceive) I have beflowed no fmall fhare of attention and labour. And by thefe additional and explanatory tracts, together with a few other mathematical tracts of my own compofition, on fubjects that have for the moit part a remarkable connection with the binomial theorem, this collection has been fwelled to fuch a fize. that that it cannot be concluded in lefs than three volumes, quarto: of which the two first are now prefented to the public.' For the fatisfaction of the reader, we shall here fubjoin, in as concife a manner as poffible, an account of the feveral. treatifes which are republithed in this collection. The first is Kepler's Chilias Logarithmorum, printed at Marpurg in 1624. II. Supplementum Chiliadis Logarithmorum, continens præcepta de eorum ufu, by the fame author; Marpurg, 1625. I. The Logarithmotechnia of Mercator, firft printed in the Philofophical Tranfactions, in 1668. IV. Exercitatio Geometrica de Maximis et Minimis, of Riccius, annexed to the preceding article in the Philofophical Tranfactions. V. An extract from the third volume of thefe Tranfactions, published in 1668, by Mr. Oldenburgh, containing a method of fquaring the hyperbola by an infinite feries of rational numbers, with its demonftration, by Lord Brouncker. VI. Another extract from the Tranfactions, No. 38, published in Auguft 1668, containing an account of Mercator's Logarithmotechnia; with another infinite feries for the quadrature of the hyperbola, by Dr. Wallis; a method of finding the fums of logarithms by the fame author; and an illuftration of Mercator's treatife by himself. These fix tracts comprehend 232 pages of the first volume of this work. In the fecond volume, the first tract is intitled, Nicolai Mercatoris Quadratura Hyperboles geometrice demonftrata, by Mr. James Gregory, extracted from the Exercitationes Geometrica, first published at London in 1668. The fecond tract, by the fame author, and taken from the fame collection, is intitled, Analogia inter Lineam Meridianam Planifphærii Nautici et Tangentes artificiales, geometricè demonftrata, &c. The third tract, intitled, Methodus componendi Tabulas Tangentium et Secantium artificialium ex Tabulis Tangentium et Secantium Naturalium, exactiffimè et minimo cum Labore, is by the fame author, and taken from the fame collection. No. IV. is an extract of a letter from the fame author to Mr. Collins, dated Feb. 15, 1670-1, first published in the Commercium Epiftolicum, &c. 1712, and containing fome infinite feries relating to the tangents and fecants of circular arcs, and to the logarithms of the ratios of fuch tangents and fecants to the radius. No. V. Extract of Newton's first epiftle to Mr. Oldenburgh, dated in June 1676, and containing a difcovery relating to logarithms. No. VI. Extract of a letter from Leibnitz to Mr. Oldenburgh, dated Auguft 1676, and containing a paffage relating to logarithms. No. VII. Extract of a fecond epiftle from Newton to Mr. Oldenburgh, dated October 16;6, and containing some difcoveries relating to logarithms. No. VIII. is the twelfth chapter of Dr. Wallis's treatife of algebra, intitled Of logarithms, their invention, and ufe; published in 1685. No. IX. Letter from Dr. Wallis to Mr. Norris concerning the collection of fecants, and the true divifion of the meridians in the fea chart, publifhed in the Philofophical Tranfactions of the year 1686. No. X. Speidell's Logarithmotechnia, London, 1688. No. XI. Halley's eafy Demonftration of the Analogy of the Logarithmic Tangents to the Meridian Line or Sum of the Secants, &c. published in the Philofophical Tranfactions for the year 1695-6. The twelfth is a moft compendious and facile method of conftructing the logarithm, &c. by the fame author, published in the Transactions for the year 1695. These twelve tracts extend through 91 pages of the fecond volume. We must now observe that it is not merely as an editor that we are to regard Baron Maferes in this publication. The volumes are enriched with many valuable tracts of his own compofition, exhibiting a very extenfive and accurate acquaintance with a variety of mathematical fubjects, and ferving to refolve difficulties and to fupply deficiencies in the difcuffion of them. In the first volume, we have remarks on the two infinite feries, which were invented by Mr. Mercator and Dr. Wallis, and applied by them to the quadrature of the hyperbola; and alfo an appendix to thefe remarks; both of which comprehend 150 pages. In the fecond volume, we have, 1. Notes on fome of the more difficult paffages of Dr. Halley's discourse on the method of constructing logarithms. 2. Appendix to Dr. Halley's tract, fhewing how to compute the logarithms of ratios in any fyltem merely by the help of Sir Ifaac Newton's binomial theorem. 3. Demonftration of this theorem in the cafe of integral powers, &c. which is followed by another demonstration publifhed by Mr. Landen in 1758 and 1764. 4. Explanation of Mr. Landen's demonftration in cafe of the fractional index 5. Difcourfe concerning the bi nomial theorem, in the cafe of fractional powers, with a demonftration of it in this cafe. 6. Difcourfe concerning Sir I. Newton's refidual theorem in the cafe of fractional powers, with a demonstration of it. 7. Method of extending Cardan's rule for refolving the cubic equation, y3+qyr, or qy+y=r, to the refolution of the cubic equation, qy-yr in particular cafes, by the help of Newton's binomial and refidual theorems. 8. Method of extending Cardan's rule for refolving the cubic equation of yqyr in one cafe to the other cafe of the fame equation under certain reftrictions, fpecified by the author. 9. Conjecture concerning the method by which Cardan's rules for refolving the cubic equation x3+qxr in all cafes, and the cubic equation x3-qxr in the first cale of it, were probably x3difcovered difcovered by Scipio Fereus of Bononia, and Nicholas Tartalea, or any other perfons who first invented them. 10. Appendix to the third tract in the preceding enumeration. The feveral articles, which we have now recited, occupy 500 pages of the fecond volume, While we congratulate the public on this new acceffion to the treasury of mathematical knowlege, we exprefs our hope that the ingenious and indefatigable editor will perfevere in completing his plan by the publication of the third volume, which his preface leads us to expect. We are not unapprized that works of this nature do not meet with the encouragement which they deferve; and that they must be undertaken by perfons who have no views to pecuniary profit, and who are contented to devote time, labour, and expence, to the improvement of useful science. To the fatisfaction and honour accruing from reflections of this kind, Baron Maferes, whether we confider him as an editor or an author, is justly entitled. Re-s. ART. V. The Theory and Practice of finding the Longitude at Sea or EVERY attempt to improve the art of navigation, by explain ing the principles and applying the obfervations on which it is founded, merits encouragement. To all commercial ftates, this art is of peculiar utility and importance; and there is no nation which is more indebted to it for fecurity and profperity than our own. To the improvement of nautical fcience, the attention of our countrymen has been laudably and fuccessfully directed; and the navigator is now in poffeffion of inftruments and tables, by the ufe of which the moft difficult problem, or that of finding the longitude, may be refolved without much labour, and with a very confiderable degree of accuracy. The treatife before us contains a comprehenfive and fatisfactory account of every thing that is neceffary to be known on this fubject; and the ingenious author, who introduces it, with modeft diffidence, to the notice of the public, is fully intitled to that approbation which he efteems highest and best reward.' his The firft of thefe volumes is divided into fix books; in the firft of which the author premifes thofe definitions and general principles that are neceffary to a proper knowlege of the fubject; in the fecond book, he explains the ftructure and use of the quadrant, fextant, and circular inftrument, in their prefent improved I |
