vented by Sir Samuel Moreland; a third, by Leibnitz; a fourth, by the Marchese Poleni; and a fifth, by Mr. Leopold. This of Gersten is founded on that of Leibnitz, but so complicated and operose as not to be entitled to any attention. The next paper is entitled, Of a New-invented Arithmetical Instrument, called a Shwan-pan, or Chinese Account Table. By Gamaliel Smethurst.* This is an instrument somewhat like the abacus of the ancients, adapted to our arithmetical notation, and conceived by Mr. Smethurst to be much superior to the Shwan-pan of the Chinese. Such inventions are not worth much attention. The common rules of arithmetic can be performed with sufficient facility without any such assistance. The next paper is entitled, An Essay towards solving a Problem in the Doctrine of Chances. By the late Rev. Mr. Bayes.† The problem is as follows: Having given the number of times an unknown event has happened and failed, to find the chance that the probability of its happening should lie somewhere between any two named degrees of probability. The solution is much too long and intricate to be of much practical utility. Dr. Price published a supplement to this paper in the succeeding volume of the Transactions.‡ The next paper is entitled, On the Theory of circulating Decimal Fractions. By John Robertson, Lib. R. S.§ The subsequent publications of Dr. Hutton, have deprived this paper of all its interest. The next paper is by the same author, and is entitled, Investigations of Twenty Cases of Compound Interest. These are investigations of cases of compound interest previously engraven on a plate, by William Jones, Esq.; and published without demonstration by Gardiner and Dodson. The last arithmetical paper we have to mention, is entitled, Some Properties of the Sum of the Divisors of Numbers. By Edward Waring, M.D.¶ Waring was one of the profoundest mathematicians of the 18th century; but the inelegance and obscurity of his writings prevented him from obtaining that reputation to which he was entitled. Except Emerson, there is scarcely any writer whose works are so revolting as those of Waring. The great elegance and admirable simplicity and order of all the writings of Euler contributed as much to give him the great celebrity which he enjoyed, as his inventive genius. 2. The papers on Logarithms, in the Philosophical Transactions, amount to 11; and some of them are of very considerable importance. The first, is an account of the Logarithmotechnia of Mercator; a book which we noticed, with due praise, while detailing the history of mathematics. * Phil. Trans. 1749. Vol. XLVI. p. 22.. Phil. Trans. 1764. Vol. XIV. p. 296. Phil, Trans. 1670. Vol. LX. p. 508. + Phil. Trans. 1763. Vol. LIII. p. 370. Phil. Trans. 1768. Vol. LVIII. p. 207. q Phil. Trans. 1788. Vol. LXXVIII. p. 388. The next paper is an excellent one by Dr. Halley, on the Method of constructing Logarithms from the Nature of Numbers, without any regard to the Hyperbola.* There is another Method of constructing Logarithms, proposed some years after, by Mr. Craig, published in a subsequent volume;† and another by Mr. Long, in a subsequent volume of the Transactions. But this method is, in some measure, mechanical; and is not to be compared to the methods of Halley and Craig. Dr. Halley published an important paper, entitled, An Easy Demonstration of the Analogy of the Logarithmic Tangents to the Meridian Line, or Sum of the Secants; with various methods for computing the same to the utmost exactness. This paper, though we mention it here, is rather connected with navigation than logarithms. The Logometria, of Cotes, deserves to be mentioned on account of its importance. It was afterwards published in the collection of Mr. Cotes's works, in 1722, by Dr. Smith, under the title of Harmonica Mensurarum. Roger Cates, Cotes was one of the most eminent mathematicians of the age in which he lived. He was born in 1682, at Burbach, in Leicestershire. He was educated at St. Paul's School, London, and afterwards at Trinity College, Cambridge; where, in 1706, he was appointed the First Professor of Astronomy and Experimental Philosophy, on the foundation of Dr. Plume, Archdeacon of Rochester. In 1713 he published a new edition of Newton's Principia; his preface to which was much admired, and procured him great reputation. He unfortunately died in the year 1716, in the thirty-fourth year of his age, to the great regret of all lovers of the sciences. Newton had so high an opinion of his knowledge and genius, that he used to say, " had Cotes lived, we should have known something." The celebrated property of the circle, which he discovered, and which is of so much importance in the integral calculus, was not known till after his death, when it was found among his papers, and made out, with great difficulty, by his relation, Dr. Smith. The next paper is An Account of the Method of protracting the Logarithmic Lines, on the common Gunter's Scale. By Mr. John Robertson. I The next paper is A Dissertation on Logarithms. By Mr. Jones.** It is remarkable for that great conciseness of expression, which distinguishes all the writings of that eminent mathematician. The view which he takes of the subject is very similar to one taken on logarithms by Euler, and which, as far as we can judge, seems to be the simplest and most perspicuous, as well as satisfactory way of viewing the subject. In a subsequent paper++ Mr. Hellins * Phil. Trans. 1695. Vol. XIX. p. 58. + Phil. Trans. 1710. Vol. XXVII. p. 191. gives Two Theorems, which he demonstrates, and which he shows to be of considerable utility in facilitating the computation of logarithms. The last paper on this subject which we think requires mentioning, is one by Mr. Nicholson, in which he gives the principles and illustration of an advantageous method of arranging the differences of logarithms, on lines graduated for the purpose of computation.* 3. There are no fewer than 15 papers on the nature of equations. But as we have noticed the most important of these already, it will be sufficient merely to refer to them in a note.† 4. On the doctrine of series, which is one of the most important branches of mathematics, and which may be considered as in some measure the original of all the modern improvements in the science, there are 13 papers. The following are their titles, with the names of the authors: Method of determining 2d, 3d, 4th, &c. Term of a Series taken in order. T. Simpson. New Method of computing the Sums of certain Series. Landen. On certain Infinite Series. Bayes. Series for computing the Ratio of the Diameter of a Circle, and its Circumference. Hutton. To find the Value of certain Infinite decreasing Series. Maseres. To find a near Value of a slowly converging Series. Maseres. New Methods of investigating the Sums of Infinite Series. Vince. On Infinite Series. Waring. On finding the Value of Algebraic Quantities, by converging Series. Waring. New Method of investigating the Sums of Infinite Series. Waring. 5. There is only one paper on interpolation; the author of which is Dr. Waring. But Newton, in his Principia, has given an excellent method, probably the first ever contrived. The subject has been much discussed by the foreign mathematicians, but Newton's method still retains its utility. 6. On the doctrine of annuities, which was in a manner first broached in * Phil. Trans. 1787. Vol. LXXVII. p. 246. + Collins. Phil. Trans. 1669. Vol. IV. p. 929.-Halley. Ibid. 1687. Vol. XVI. p. 335 and 387; and Vol. XVIII. p. 136.-Colson. Ibid. 1707. Vol. XXV. p. 2353.-Demoivre. Ibid. Vol. XXV. p. 2368.—Maclaurin. Ibid. 1726. Vol. XXXIV. p. 104.-Milner. Ibid. 1778. Vol. LXVIII. p. 380. -Maseres. Ibid. Vol. LXVIII. p. 902.; and Vol. LXX. p. 85 and 221.-Waring. Ibid. 1779. Vol. LXIX. p. 86.-Lord Stanhope. Ibid. 1781. Vol. LXXI. p. 195.-Wales. Ibid. 1781. Vol. LXXI. p. 454.-Hellins. Ibid. 1782. Vol. LXXII. p. 417.-Wood. Ibid. 1798. Vol. LXXXVII. p. 369. Wilson. Ibid. 1799. Vol. LXXXIX. p. 265. this country, and which has always been cultivated with much assiduity, there are eight papers. The following are their titles: Value of an Annuity for Life, and Probability of Survivorship. Dodson. Method of calculating Reversions depending on Survivorship. Price. Theorems for Annuities. Price. On the Probabilities of Survivorships, &c. Morgan. Determination of Contingent Reversions. Morgan. On determining the real Probabilities of Life, &c. Morgan. 7. There are three papers on the quadrature of the circle; not reckoning the account of James Gregory's book on the subject, and his paper in answer to the objections of Huygens. The first paper is by Leibnitz; the second is by Dr. Hutton, and has been already mentioned under the head of Series; the third paper is by Mr. Hellins; and is an improvement of Halley's method of computing the quadrature of the circle. 8. Upon the conic sections, and other curves of a higher order, we have a considerable number of papers. The following are the chief: New Properties in Conic Sections. Waring. Length of the Arc of a Conic Hyperbola found. Landen. New Quadrature of the Hyperbola. Demoivre. Hyperbolical Cylindroid. Wren. Testudo Veliformis Quadrabilis. David Gregory. Synchronism of the Vibrations in a Cycloid. Quadrable Cycloidal Spaces. Wallis. Quadrature of a Portion of an Epicycloid. Caswell. Method of Measuring all Cycloids and Epicycloids. Halley. History of the Cycloid. Wallis. Properties of the Catenaria. Gregory. Curve, called Cardioid. Castillion. Curve of Swiftest Descent. Sault and Craig, and Machin, in three different papers. The problem, as first proposed by Bernoulli, is likewise resolved by Sir Isaac Newton. Quadrature of the Logarithmic Curve. Craig. Quadrature of the Lunula of Hippocrates. Pertus and Gregory. Dimension of the Solids, produced by Hippocrates' Lunula. Easy Method of measuring Curvelinear Figures. Wallis. Demoivre, Of squaring some Kinds of Curves. Demoivre. General Method of determining the Quadrature of Figures. Craig, Quadrature of the Foliate, a Curve of the Third Order. Demoivre. General Method of describing Curves by the Intersection of Right Lines. Brakenridge. Description of Curve Lines. Maclaurin. On finding Curve Lines from the Properties of the. Variation of Curvature. Two Species of Lines of the Third Order, not mentioned by Newton nor Stirling. Stone. Length of Curve Lines. Craig. Measure of Curves, and their Construction. Maclaurin. Tangents to Curves. Sluse. Tangents of Curves, deduced from the Theory of Maxima & Minima. Ditton. Collection of Secants. Wallis. 9. On Plane and Spherical Trigonometry there are few papers in the Transactions; unless we were to include, under that title, some of those mentioned under the preceding heads, which might certainly be done without impropriety. The following are the only papers on that subject which I at present recollect: Spherical Trigonometry reduced to Plane. Blake. Trigonometry abridged. Murdoch. Calculations in Spherical Trigonometry abridged. Lyons. 10. We shall now give a list of a few papers, which could not, with propriety, be reduced under any of the preceding heads. Equations for exhibiting the Resolution of Goniometrical Lines. Jones. Locus of three or four Lines, famous among the Ancient Geometricians. Pemberton. On the Bases of Cells, where Bees lodge their Honey. Maclaurin. The above lists may be considered as exhibiting the names of all the mathematical papers contained in the Philosophical Transactions, except a few papers on the fluctionary calculus, and subjects connected with it; the titles of which could not have been rendered intelligible, without entering into further details than is consistent with this work. Here then we take our leave of this subject; upon which it would have been much easier to have extended our observations to a far greater length, than to have confined ourselves to the short space which our account of Mathematics occupies. But we did not consider ourselves as at liberty to sacrifice the different branches of Mechanical Philosophy and Chemistry, which, to most of our readers will probably be more interesting than dry details respecting such abstract and difficult subjects. |