1776 Mathematical Correfpondence. 153 We were favoured with answers to this question by the Propofer, Mr. Merritt, Mr. Robbins, Mr. Keech, and others. [47.] QUESTION II. Anfwered by Mr. Joshua Merritt. A B E G If the given bafe be bifected in E, and from O the center of the infcribed circle OF be let fall to the bafe AB then EF will be = to the given difference of the fides. (Vide p. 97 of the London Magazine for Feb. 1776) Again, if OB be joined cutting the circumference of the infcribed circle in D, then hall BD be the given diftance from the angle B. Now if OF be produced to G fo that FGDB and GB be joined, the triangle OGB will be ifofceles; and the angle OGB → the angle OBG (by Euc. 5. 1.) whence the following conftruction is evident. Conf. Take AB the bafe, which bifect in E, take EF the given difference of the fides, at F erect an indefinite L FO which produce below AB to G, fo that FG the given diftance from the angle at the bafe; then join GB and draw BO cutting the indefinite 1, in O making an angle with GB to BGO, also from O with OF radius defcribe a circle; lafly from the points A and B draw AC and BC to touch the circle, and meeting each other in C, and ACB is the triangle required. Elegant conftructions of this question were fent by Mr. Robbins, the proposer, Mr. Bonnycastle, and others. [48.] QUESTION III. Anfwered by Mr. Ifaac Dalby. = W B P N 171 Projection. Let GWA be the primitive, and D and NO the two leffer circles. In any pofition, as NO, let the given arc be drawn within one of the leffer circles, and let a leffer parallel CQe to mNO be drawn to touch the arc NO; draw the great circles AE, HG parallel to mNO, and D refpectively; then having H drawn a great circle WF making the angles ASF, HVW the diftances of the parallels AE, Ce; and HG, Drefpectively, that great circle will touch the leffer D and have the intercepted arc nr the arc NO; alfo if another great circle BI be drawn to make the angle ERI ASF, and HKB = HVW, it will answer the conditions of the problem. The truth of this conftruction is evident, because if a great circle has a leffer one drawn parallel to it at a given diftance, and another great circle be drawn to touch the leffer, the angle made by the great circles will be the given diftance of the parallels; hence it follows by the conftruction, that the great circles WF, BI touch the leffer ones D, C, and fince CQe and m NO are parallels, it requires no demonstration to prove that all the arcs of great circles, as NO, nr, my, intetcepted by the outer m NO, and touching the inner CO are equal to each other. If it was required to draw a great circle through two leffer ones fo as to have given intercepted arcs, the projection will be the fame with the foregoing. The Propofer alfo favoured us with a folution. NEW MATHEMATICAL QUESTIONS. [52] QUESTION I. By Cleonicus. IN Hawney's Menfuration it is faid, that if a chord AC of a given fegment of a circle ABC be divided into four equal parts, AD being equal to one of them, and if AD be applyed to the arch from C to E and DE be joined, DE will be equal to half the arch ABC: a demonftration of the truth or fallacy of this is required. March 1776. X [53.] QUES Review of New Publications. March 754 [53.] QUESTION II. By Mr. Robert Phillips, of Saint Agnes in Cornwall. IN a right angled triangle there is given, the fum of the fides and the fut of the two lines drawn from the acute angles to the center of the infcribed circle; to construct the triangle. [54.] QUESTION III. By Theon. IF a ball be laid on a given point of the curve of a given ellipfe, whofe tranfverfe diameter is parallel and conjugate perpendicular to the horizon, and left to defcend by its own gravity; it is required to determine the point where it will quit the curve. N. B. Our correspondents are intreated to fend their favours by the tenth of each month. TH An Impartial Review of New Publications. ARTICLE XXXI. HE Hiftory of the Decline and Fall of the Roman Empire. By Edward Gibbon, Efq. Vol. I. 11. 2s. Cadell. We shall give the author's plan in his own words, "The memorable feries of revolutions, which, in the course of about thirteen centuries, gradually undermined, and at length deftroyed the folid fabric of Roman greatnefs, may, with fome propriety, be divided into the three following periods. "I. The first of thefe periods may be traced from the age of Trajan and the Antonines, when the Roman monarchy having attained its full ftrength and maturity, began to verge towards its decline; and will extend to the fubverfion of the western empire, by the barbarians of Germany and Scythia, the rude ancestors of the most polifhed nations of modern Europe. This extraordinary revolution, which fubjected Rome to the power of a Gothic conqueror, was completed about the beginning of the fixth century. II. The fecond period of the decline and fall of Rome, may be fuppofed to commence with the reign of Juftinian, who by his laws, as well as by his victories, restored a tranfient fplendor to the eastern empire. It will comprehend the invafion of Italy by the Lombards; the conqueft of the Afiatic and African provinces by the Arabs, who embraced the religion of Mahomet: the revolt of the Roman people against the feeble princes of Conftantinople, and the elevation of Charlemagne, who, in the year eight hundred, eftablished the fecond or German empire of the weft. III. The laft and longeft of thefe periods includes about feven centuries and a half; from the revival of the western empire, till the taking of Conftantinople by the Turks, and the extinction of a degenerate race of princes, who continued to affume the titles of Cæfar and Auguftus, after their dominions were contracted to the limits of a fingle city; in which the language, as well as the manners of the ancient Romans, had been long fince forgotten. The writer who fhould undertake to relate the events of this period, would find himself obliged to enter into the general hiftory of the Crufades, as far as they contributed to the ruin of the Greek empire; and he would scarcely be able to reftrain his curiofity from making fome enquiry into the fate of the city of Rome, during the darkness and confufion of the middle ages." Mr. Gibbon hath, in the prefent volume, treated of the first period, beginning with the age of Trajan and the Antonines, and brought it down only to that of Conftantine. His plan is very extenfive, and his views of the Roman history more comprehenfive and accurate than any hiftorian who have preceded him. His ftyle alfo is manly, perfpicuous, and in a degree elegant. His obfervations are judicious, and feem strongly pointed to the decline and fall of the British, as well as Roman, Empire. The fame caufes which brought on the ruin of the last, are too vifible in the other, corruption, luxury, ambition, and ftanding armies. The following are his remarks on Auguftus, worthy the attention of our fenators. "The tender refpect of Auguftus for free conftitution which he had deftroyed, can only be explained by an attentive confideration of the character of that fubtile tyrant. A cool head, an unfeeling heart, and a cowardly difpofition, prompted him, at the age of nineteen, to affume the mafk of hypocrify, which he never afterwards laid afide. With the fame hand, and probably with the fame temper, he figned the profcription of Cicero, and the pardon of Cinna. His virtues and even his vices were artificial and according to the various dictates of his intereft, he was at firft the enemy, and at laft the father of the Roman world. he framed the artful fyftem of the imperial authority, his moderation was infpired by his fears. He wished to deceive the people When 1776. Gibbon's Decline of the Roman Empire. 155 by an image of civil liberty, and the armies dreaded their calmer moments of reflection. by an image of civil government. I. The death of Cæfar was ever before his eyes. He had lavished wealth and honours on his adherents: but the most favoursed friends of his uncle were in the number of the confpirators. The fidelity of the legions might defend his authority against open rebellion; but their vigilance could not fecure his perfon from the dagger of a determined republican; and the Romans who rewered the memory of Brutus, would applaud the imitation of his virtue. Cæfar had provoked his fate, as much by the oftentation of his power, as by his power itself. The conful or the tribune might have reigned in peace. The title of king had armed the Romans against his life. Auguftus was fenfible that mankind is governed by names, nor was he deceived in his expectation, that the fenate and people would fubmit to flavery, provided they were refpectfully affured that they still enjoyed their ancient freedom. A feeble fenate and enervated people cheerfully acquiefced in the pleafing illufion, as long as it was fupported by the virtue, or by even the prudence of the fucceffors of Auguftus. It was a motive of felf prefervation, not a principle of liberty, that animated the confpirators against Caligula, Nero, and Domitian. They attacked the perfon of the tyrant, without aiming their blow at the authority of the emperor. There appears, indeed, one memorable occafion, in which the fenate, after feventy years of patience, made an ineffectual attempt to reaffume its long forgotten rights. When the throne was vacant by the murder of Caligula, the confuls convoked that affembly in the capital, condemned the memory of the Cæfars, gave the watch word Liberty to the few cohorts who faintly adhered to their ftandard, and during eight and forty hours, acted as the independent chiefs of a tree commonwealth. But while they deliberated, the prætorian guards had refolwed. The ftupid Claudius, brother of Germanicus, was already in their camp, invefted with the imperial purple, and prepared to ppport his election by arms. The dream of liberty was at an end; and the fenate awoke to all the horrors of inevitable fervitude. Deferted by the people, and threatened by a military force, that feeble affembly was compelled to ratify the choice of the prætorians, and to embrace the benefit of an amnefty, which Claudius had the prudence to offer, and the generofity to obferve. II. The infolence of the armies infpired Auguftus with fears of a ftill more alarming nature. The defpair of the citizens could only attempt, what the power of the foldiers was, at any time, able to execute. How precarious was his own authority over men whom he had taught to violate every focial duty! he had heard their feditious clamours; he One revolution had been purchased by immenfe rewards; but a fecond revolution might double thofe rewards. The troops profeffed the fondeft attachment to the house of Cæfar; but the attachments of the multitude are capricious and inconftant. Auguf tus fummoned to his aid, whatever remained in those fierce minds, of Roman prejudices; enforced the rigour of difcipline by the fanction of law; and interpofing the ma jefty of the fenate, between the emperor and the army, boldly claimed their allegiance, as the first magiftrate of the republic." XXXII. Liberal Opinions upon Animals, Man, and Providence, in which are introduced Anecdotes of a Gentleman, addressed to the Rt. Hon. Lady Cb-. By Courtney Melmoth, 2 vols. 55. Robinfon. The author's defign in thefe volumes is to vindicate the ways of God to man, and even to brutes---aud to prove that amidst the apparent confufion in providential difpenfa-; tions To be good, is to be happy. The confufion is regular, and Mr. Melmoth difplays great humanity, as well as a good knowledge of mankind. The fubject is alfo enlivened with fome agreeable poetical pieces, and inftructive anecdotes. Our author is fo great a friend to brutes, as to efpoufe the opinion of their immortality." Nothing (fays he) could give a greater fhock to my fenfibility, than the horrid idea of univerfal annihilation prevailing over the animal world. Muft my dear Tabythyetta-my demure Grimalcana-my merry Scugypugiffa (you will pardon me, madam, for Italianizing their names, there is fomething fo dreadfully dull and mechanic in the found of an English appellation) muft all thefe, with that great travelling Triplea, fink into nothingnefs! into oblivion! into dit! Oh horrible, horrible, moft horrible !" Our author fays, that among the facred writers, Mofes and Solomon have leaned much in favour of animals immortality; but the paffages are not quoted-among the moderns, he ranks Mr. Locke, Dr. Hildrop, Soame Jennings, and many others-even French, Spanish, and Roman authors have leaned to that fide alfo-fo that he thinks there is fufficient fanction for his entering the lift, as the champion of millions of ufeful, beautiful, and innocent beings. XXXHI. Annals of Scotland, from the Acceffion of Malcolm III. firnamed Canmore to the Acceffion of Robert I. By Sir David Dal rymple. 155. Murray. The learned author commences his history with Malcolm III. because all before that period, in Scotland, is involved in obfcurity and fable. If thefe annals are approved of, and the author's health permits, he pro X 2 pofes 156 Curious Particulars concerning Macbeth. Potes to continue them to the restoration of James I. No perfon appears more capable of elucidating history-his induftry, erudition and critical difcernment are very manifeft,and the prefent werk will be read both with pleafure and profit. We have feltted the fallowing account of thofe perfons on whom Shakespeare founded his famous tragedy of M'Beth, with the learned annotations, as a Specimen. "Malcolm II King of Scotland had a daughter, Beatrice, the mother of Duncan, Fordun. iv. 49. In 1034, Duncan fucceeded his grandfather Malcolm. In 10:9 he was affeinated by M'Beth. * By his wife, the fifter † of Siward, Earl of Northumberland, he left two fons, Malcolm, firnamed Canmore, and Donald, fire named Bine .-Fordun, iv. 49. M'Beth expelled the fons of Duncan, and ufurped the Scottish throne. Malcolm fought refuge in Cumberland, Donald, in the Hebrides.-Fordun, iv. 51. T March When Edward the Confeffor fucceeded to the Crown of England [1043] Earl Siward placed Malcolm under his protection. Malcolm remained long at his court, an honoure able and neglected exile,-Fordun, iv. 54. The partizans of Malcolm often attempted to procure his restoration; but their efforts, feeble and ill-concerted, only ferved to efta blish the dominion of the ufurper.-Fordun, iv. 7. At length, M'Duff thane of Fife, excited a formidable revolt in Scotland, while Siward, with the approbation of his fovereign, led the Northumbrians to the aid of his nephew Malcolm. He lived not to fee the event of this generous enterprize Fordun, v. 7.—Cbr. Lax. 169. M'Beth retreated to the faftaeffes of the North, and protracted the war. His people forfook his ftandard. Malcolm attacked him at Lunfanan ** in Aberdeenshire: abandoned by his few remaining followers, M' Beth fell [5th December 1056] Fordun, v.7. The At Inverness according to later Hiftorians, who follow Boece. The registry of the priory of St. Andrews fays, in Bothgeuanan. Fordun fays, I. iv. c. 49. that, being wounded, be was. conveyed to Elgin, and died there. The word Bothgauanan means, in Gaelic, the fmith's dweling. It is probable, that the affaffins lay in ambush, and murdered bim, at a fmith's boufe in the neighbourhood of Elgin." Fordun calls ber configuinea comitis. Other biftorians call her bis daughter. Cean-more, or great-head, according to the rude diflinɛtions used in those times, Bane, white, or of a fair complexion. Our biftorians relate, that M`Duff, in an interview with Malcolm, propofed the plan of bis refloration, but that the young prince, fufpecting the fidelity of M'Duff, artfully pretended that he knew himself to be unworthy of a crown. He urged, that he was libidinous, a thief, and void of faith. On this falfe confidence, a moft abfurd converfation enfued, according to Fordun, I. v. c. 1. 6. Buchanan bas polished the narrative, and bestowed a plausible appearance on the fable; 1. vii. p. 114. The Saxon chronicle places his death in 1055; p. 169. Ingalphus, in 1056; p. 66. There is a curious paffage concerning bim in Leland's Collectanea, t. 1. p. 529. "Sivard fent bis funne to warre in Scotland, wher ke dyid of the flux; where be after toke the fame difeafe, ard dyid of it. But be, much detefting to dy like a cow of the flux, cauffid himself to be armid at all peaces, and dyed yn bis armure." Brompton, p. 946. relates the dying words of Siward in a bombaft file, which be mistook for eloquence. The account which Brompton gives of Siward is ridiculous and contradi&tory. It ends with faying, that Sirvard put King Duncan in poffeffion of Scotland. In Aberdeenshire, two miles north west of the village of Kincardino Neil. Fuft by the paris-church of Lunfanan, there is a valley where the veftiges of an ancient fortress are still to be discerned. It has been of an chlong figure; in length, near an bundred yards, and twenty yards in breadth. Abrook, which waters the valley, bas been led round it. As no remains of buildings are to be feen, it is probable that the fortress was composed of timber and fod. In this Jilitary place, we may conjecture that M'Berk fought an afylum. ++ We know few particulars of the real biftory of M'Beth. Boece fays, that his mother svas Deada, a daughter of Malcolm II. and, confequently, that be was the cousin german of Durcan; 1. xii. fol. 246. b. Fordun feems to have been ignorant of bis connection with the Royal Family. He terms kim fimply, A1 Beth the son of Finele (or Finlay;) 1. iv. c. 49. Race bas exerted all his inventive powers in delineating the character and biftory of M'Betb. The genius of Shakespeare gave jach firength of colouring to the portrait, that the fictions of Riece offumed the form of Litical realities. The wetard filers, the wood of Burnam advancing to Durfinnan, the prophecy that I Beth fould never be overcome by any one born of a woman, are incidents which the lift age devoutly believed. Buchanan artfully foftened the im-. probabilities of the tale of the weird Jifiers, or the three witches: and foftens the apparition inta a dream. Buece records the laws of M Betb. They are a palpaple forgery. If Brece bimJelf was the farger, they fhew bis ignorance of history and manners. See 1. xii. p. 250. b. There is a fingular poflage concerning M`Betb in Florence of Worcefler, p. 626. "Anno 1950, Rex $otorum Af Bethud Rome argentum fpargendo duty buit. "Fordan jimply adds, 1776. Review of New Publications. The kindred of M'Beth placed a relation of his on the throne 11. No party efpoufed the cause of this pageant monarch. Malcolm foon discovered his Jurking place, and Lew him [at Effe in Strathbolgie, 3d April 1057.] - Fordun, v. S. Chr. Melrof. 158. Malcolm was crowned at Scone [on the festival of St. Mark, 25th April 1057.] Fordun, V.9. The patriot, who reftored Malcolm to the throne of his ancestors, demanded no reward in titles of dignity, penfions, or grants of crown-lands. The privileges which M'Duff fought, and the king beftowed, were, 1. That he, and his fucceffors, lords of Fife, fhould have the right of placing the kings of Scotland on the throne, at their coronation. 2. That they should lead the van of the Scottish armies, whenever the royal banner was displayed. 3. That, if he, or any of his kindred, committed flaughter of fuddenty, they fhould have a peculiar fanctuary, and retain remiffion, on payment of an atonement in money." XXXIV. Obfervations on the Nature of civil Liberty, the Principles of Government, and the Juftice and Policy of the War with America, to which is added an Appendix containing a State of the national Debt, &c. By Richard Price, D. D. F. R. S. 23. Cadell. The learned author hath here taken the liberty of examining public measures with great freedom and fpirit, which, happily for this kingdom, every perfon in it at prefent enjoys. In our Laft magazine, our readers were prefented with the fubftance of the appendix on the national debt.-we shall therefore only give them the Doctor's idea of liber ty in general, without having correct ideas, of which, none can judge properly of the queftion between adminiftration and the American colonies. 157 we fies the power of exercising, without moleftation, that mode of religion which think beft; or of making the decifions of our own confciences, refpecting religious truth, the rule of our conduct, and not any of the decifions of others.-In like manner; civil liberty is the power of a civil fociety or ftate to govern itself by its own difcretion; or by laws of its own making, without being fubject to any foreign difcretion, or to the impofitions of any extraneous will or power. It should be obferved, that, according to thefe definitions of the different kinds of liberty, there is one general idea, that runs through them all; I mean, the idea of selfdirection, or felf-government.-Did our volitions originate not with ourselves, but with fome caufe over which we have no power; or were we under a neceffity of al-' ways following fome will different from our own, we should want phyfical liberty. In like manner; he whofe perceptions of moral obligation are controuled by his paffions has lost his moral liberty; and the most common language applied to him is, that he wants felf-government. He likewife who, in religion, cannot govern himself by his convictions of religious duty, but is obliged to receive formularies of faith, and to practife modes of worship impofed upon him by others, wants religious liberty.-and the community also that is governed, not by itself, but by fome will inde pendent of it, and over which it has no controul, wants civil liberty. In all these cafes there is a force which ftands oppofed to the agent's own will; and which, as far as it operates, produces fervitude.-In the firft cafe, this force is incompatible with the very idea of voluntary motion; and the fubject of it is a mere paffive inftrument which never acts, but is always acted upon.-In the fecond cafe, this force is the influence of paffion getting the better of reafon; or the brute overpowering and conquering the will of theman. In the third cafe; it is human authority in religion requiring conformity to particular modes of faith and worship, and fuperfeding private judgment. -And in the laft cafe, it is any will diftin& from that of the majority of a community, which claims a power of making laws for it, and difpofing of its property. "By phyfical liberty I mean that principle, of fpontaneity, or felf-determination, which conflitates us agents; or which gives us a command over our actions, rendering them properly curs, and not effects of the opera. tion of any foreign cause,-moral liberty is the power of following, in all circumstances, our fenfe of right and wrong; or of acting. in conformity to our reflecting and moral principles, without being controuled by any contrary principles.-Religious liberty figni"pauperibus;" 1. v. c. 9. because that word follows in the text, Pf. 112. From the words of Fordun, Goodall draws this notable inference, « Machebeda Romam profectus; "Index ad ForThus, from Fordun, and bis publisher, we learn," that M'Beth went to Rome, and bere diftributed alms to the poor;" cubertas the original infinuated, "that M Beth bribed the court of Rome." dun. This 11 Fordun calls Lims "nomine Lulach, ignomine fatuus;" 1. v. c. 8. I fufpect that Lutach was rather bis vulgar firname, than his name. In the ancient Gaelic, Luailach fignifies Mimicus, Gefticulofus, Lhuyd, Archiel. This may not unaptly be tranflated, the fool, in a fenfe familiar to our forefathers. Fordun ignorant of the Gaelic language bimfelf, and bearing this mock-monarch termed Lulach, and the fool, bas fuppofed the one to be bis name, and the other bis vulgar firname. I call Lulach a relation of M'Beth, for it is not certain that be was bis fon. |