be a tranflation of that admirable work, with a copious commentary and numerous explanatory notes. The Litchfield Society have fhewn themfelves qualified for the tafk, and it is to be hoped that they will not difcontinue their labours, but proceed to render the science every material fervice in their power. ART. VIH. PHILOSOPHICAL TRANSACTIONS of the Royal Society of London, Vol. LXXVII. For the Year 1787. Part II. 4to. 8s. Davis. 1787. MATHEMATICAL and ASTRONOMICAL Papers. The Principles and Ilußration of an advantageous Method of arranging the Dfferences of Logarithms on Lines graduated for the Purpose of Compu'ation. By William Nicholson. 1 NSTRUMENTAL Arithmetic has been of fuch great utility in the ordinary employments of life, and fo cafy and practicable as to be managed by people unfkilled in the principles on which it is founded, that every attempt to improve it is laudable: but, when these attempts have been fuccefsful, and when a real improvement has been made, peculiar praife is due to the perfon who thus confers benefits on mankind. The great univerfality and extenfive ufe of Gunter's fcale, and the eafe with which, in two minutes, it truly folves the moft intricate problems in menfuration, gauging, furveying, navigation, and many other arts, renders it an inftrument of great importance in common life. That it has a defect cannot be denied, viz. that it is not, in its prefent ftate, applicable to very large numbers, on account of the fmallness of the divifions. Mr. Nicholfon's ingenious contrivance fupplies that defect; and he has, in this paper, fhewn how the lines may be arranged on a twelve-inch fcale, fo that the divifions of the logarithmic lines may, in thatlength, be equal to those which, in the ufual way, would require a fcale of twenty feet. His method is to divide the logarithmic line, twelve feet long, into ten equal parts, and to place thefe parts, parallel to each other, on a twelve inch fcale. The intervals between the antecedents and confequents are taken by a beam compafs adapted to the fcale. The principles on which the inftrument is conftructed, are demonstrated from the properties of two or more geometrical feries', which have the fame common ratio as a, an, an2, an3, an1, an3, &c. b, bn, bn, bn', bn', bn3, &c. The ratio of any term in one feries, to its correfponding one in the other, will be conftant. Thus a: ban: bn, and :: an3: bn3, &c. or diagonally a bn' an3: bns, &c. Having demonftrated thefe properties, Mr. Nicholfon proceeds to his principal propofition, which is as follows: If the differences of the logarithms of numbers be laid in order upon an arrangement of equidiftant parallel right lines, in fuch a manner as that a right line, drawn across the whole, fhall interfect it at divifions, which denote numbers in geometrical progreffion; then, from the condition of the arrangement and the property of the logarithmic line, it follows, 1ft, That every right line, fo drawn, will, by its interfections, indicate a geometrical feries of numbers. zdly, That fuch feries, as are fo indicated by parallel right lines, will have the fame common ratio: and, 3dly, That the feries thus indicated by two parallel right lines, fuppofed to move laterally without changing their mutual distance or parallelifm to themfelves, will have each the fame common ratio, and in all pairs of feries indicated by fuch two lines, the ratio between an antecedent on one parallel, and the oppofite term on the other, taken as a confequent, will be conftant. If therefore an antecedent and confequent be given, it will be poffible to find both on the arrangement, and to draw two parallel lines, one over each number; and if the lines be then fuppofed to move, without changing either their diftance or abfolute direction, fo that the line, which before marked an antecedent, may now mark a new antecedent, the other will mark a number, at the fame relative pofition and diftance, which fhall be the confequent to this laft antecedent after the fame ratio.' Thefe principles are fufficiently obvious to a mathematician without demonftration; and on thefe Mr. N. has conftructed the fcale; which, as we faid above, confifts of ten parallel logarithmic lines, each being the tenth part of Gunter's line: fo that the first line ends at the divifion 1.2589&c. i. e. 10%, the fecond at 1.5848&c. or 1c; the third at 1.9952 &c. or 10%, and fo on to the tenth, which ends at 10. Two moveable straight edges are fo contrived, as to cut all these ten parallel lines at any diftance required, one being fet to cut the antecedent, and the other the confequent, of any ratio; and, being fixed in a frame, fo that their parallelifm may be preserved, they may be moved together, until one of them cuts a new antecedent; the fecond will therefore cut the confequent corresponding to it. The fimplicity of the inftrument renders its ufe extremely eafy, and the largeness of the divisions gives it an advantage fo fuperior to the fliding rules now in ufe, that we think it will be univerfally adopted. Mr. Nicholfon does not confine the arrangement of the parallel lines to a straight direction. They may be arranged in parallel circles, and, inftead of the two parallel edges to cut them, two radii, moving on a centre, may be fubftituted in their places. This laft is a moft admirable contrivance, as it unites, in one inftrument, both the excellencies of Gunter's line and fector, and has this material advantage, that it very much magnifies the divifions at the extremity of the line of fines, at the beginning of of the line of versed fines, and at the middle of the line of tangents. Were we to enlarge on all these particulars, we might fill many pages: we fhall therefore refer our readers to the volume before us, not doubting that they will applaud Mr. Nicholson's ingenuity. On the Preceffion of the Equinoxes. By the Rev. Samuel Vince, M. A. F. R. S. Though the annual quantity of the preceffion has been accurately determined, both by calculation and obfervation, yet the difficulty and abftrufenefs of the former, and the nicety required in the latter method, fufficiently juftify Mr. Vince in attempting to obtain a more fimple and eafy folution of the problem than hath been hitherto given. The principles on which he proceeds are juft, and his inveftigation is ingenious; but the data, whence the conclufion is drawn, are uncertain. This indeed has been the cafe in every folution, and the estimation of the quantity of the preceffion will remain fubject to much uncertainty, until the figure of the earth, the nutation of the axis, and whether the earth is uniformly denfe in all its parts, be accurately known. Mr. Vince acknowledges thefe defiderata, and he has fully accomplished his defign in giving, from the data in his poffeffion, an eafy and fimple method of determining the preceffion of the equinoxes, as far as it depends on the action of the fun's attraction. Account of an Obfervation of the right Afcenfion and Declination of Mercury out of the Meridian, near his greatest Elongation, September 1786. By John Smeaton, F. R. S. Mr. Smeaton here describes an inftrument for making fuch an obfervation on a planet, out of the meridian, as will afford data for computing the planet's right afcenfion and declination. The inftrument confifts of a micrometer telescope, fo contrived as to be manageable on an equatorial motion. The Author's ingenuity, with refpect to mechanical inventions, has long been the fubject of public approbation; and in this inftance he has given a fresh proof how defervedly that approbation has been acquired. The excellency of every mechanical invention depends on the affirmative anfwer to this obvious queftion: Does it accomplish the purpose, to which it is fubfervient, better than the contrivances that have been in common ufe before it ? If it were impoffible to obferve Mercury on the meridian, or if the obfervation made out of the meridian afcertain his right afcenfion and declination more accurately, or with lefs calculation, than the former; then Mr. Smeaton's method must be preferable to that which is commonly used. But as the planet may be be seen on the meridian, and his right afcenfion and declination are deducible, from fuch an obfervation, eafier and more accurately than from any other, it appears that our ingenious Author has exerted his mechanical talents, in this cafe, without rendering any material service to the science, or facilitating the labours of the practical aftronomer. The obfervation which Mr. Smeaton here records, fixes the right afcenfion of Mercury at 163° 59′ 21′′, and his declination 7° 44′ 25′′ north, on September 23, 1786, 5h 12′ 35′′ mean time. We fear that this fingle obfervation will not tend much to correct the theory of Mercury's orbit. If Mr. Smeaton continues his obfervations, and favours the Public with ten or twelve places of Mercury, at or about the times of 3 or 4 elongations, the science of aftronomy would reap confiderable advantage; for there is great reason to fufpect that our present theory of that planet is imperfect. We would certainly recommend the meridian obfervation, fince it is much more eafy to be made, and would determine the right afcenfion and declination of the planet, with at leaft five times lefs calculation, than is neceffary to be used in the reduction of the exmeridian observation. A Supplement to Major-general Roy's Account of the Mode propo fed to be followed in determining the relative Situation of the Royal Obfervatories of Greenwich and Paris. General Roy here corrects an error in his account of the propofed measurement published in the firft part of the prefent volume. He copied it from M. Bouguer, and remarked, that there appeared fome inconfiftency in the French philofopher's method for obtaining the length of a degree of a great circle perpendicular to the meridian. Dr. Mafkelyne pointed out the fource of the error, and obligingly communicated it to General Roy, who took the firft opportunity of giving it, together with fome other corrections, to the Public, through the fame channel in which the account was originally conveyed. [The Medical and Anatomical Papers in our next.] * Of which we gave an account in our Review for September laft, page 180. ART. IX. Supplement to the Letters written by the late Philip Dormer Stanhope, Earl of Chesterfield, to his Son, Philip Stanhope, Efq. late Envoy Extraordinary at the Court of Drefden. 4to. 55. fewed. Dodfley. 1787. F the former feries of Lord Chesterfield's Letters to his Son, we gave our opinion, at large, in the fiftieth and fifty-first volumes of our Review. Of this Supplement to that publication, the following account is given by the Editor: 10 • Soon Soon after the publication of the late Lord Chesterfield's Letters to his Son, the additional ones contained in the prefent collection were recovered by Mrs. Stanhope. That he had a defign to publish them appears from the following Preface, which was found with the Letters, foon after Mrs. Stanhope's decease, in her own handwriting: PREFACE. By Mrs. E. Stanhope. "The favourable reception of the late Earl of Chesterfield's Letters to his Son, encourages the Editor to hope that a completion of them will be acceptable to the Public. It has often been regretted that feveral of the first letters were wanting; fome in particular, which form an epitome of English hiftory, and others on geographical fubjects. Most of thefe deficiencies are fupplied in the prefent publication. "In order to prove the authenticity of the letters in this collection, as well as to teftify her gratitude, the Editor begs leave to inform the Public, that the originals, now in her poffeffion, were moft generously tranfmitted to her by Sir William Forbes, whofe father inherited them from the late Sir Andrew Mitchell. How they came. into Sir Andrew's poffeffion is not known; probably they were left. with him at Berlin, by the late Mr. Philip Stanhope. "To the letters written to Mr. Stanhope in his juvenile days, are added fome of a more recent date. One of thefe, containing the character of the late Duke of Ormond, and fome circumstances relative to that nobleman's impeachment, will, perhaps, be deemed interesting." Of the authenticity of thefe Supplemental Letters, there can be no doubt of their importance, little can be faid. Some of them are not written in Lord C.'s beft manner. He would (but there is no reading in the grave!) be forry to fee them in print, with fo few of the graces of compofition. Yet, as he did not intend any part of his fond correfpondence with his dear little boy" for public perufal, the want of a laboured correctness, or the highest polish of ftyle, ought not to bring down the smallest reflection on his Lordship's literary fame; for he was, undoubtedly, one of the moft polite writers of the age: and we juftly apply to him what Pope faid of Voiture: "His eafy art did happy nature seem ; ART. X. A Provincial Glossary, with a Collection of local Proverbs, and popular Superftitions. By Francis Grofe, Efq. 8vo. 5s. Boards. Hooper. 1787. TH HE Public are indebted to Captain Grofe for feveral ingenious and elegant works. They are now to thank him for one which will, in part, be useful: a provincial glossary will no doubt be acceptable to all perfons defirous of understanding our ancient writers. Divers partial collections,' fays the publisher, ⚫ have been occafionally made, all which have been well received, and |