children. Either then the British administration were too generous to their refpectable and compaflionate allies, or thefe good creatures eftimated the blood of females and of children at prices more exorbitant than they ufually demanded.'

On the negociations for a change of ministry we find some curious, and to us new information, but it cannot be easily detached from the body of the narrative, and we must therefore refer our readers to the work itself. On the contractor's bill our author thus expreffes himself :

While the affairs of Ireland were in this happy train of adjustment, the plans of reformation and economy which had been recommended by the miniftry, were profecuted with vigour in the British parlia ment. The bills for excluding contractors from feats in the house of commons, and incapacitating revenue-officers from voting at elections for members of parliament, were paffed, with a feeble oppofition from lord Mansfield, and a vexatious and frivolous feries of objections from the chancellor.

In the course of the debate on the contractor's bill, the chancel lor ftyled it "a puny regulation, only calculated to deceive and betray the people."-On very different principles from those of the noble lord, all good patriots muft fee, that it is indeed a puny regulation. Greatly as the principle of the bill must be approved by all honeft men, it cannot but afford matter of furprise, that its probable inefficiency fhould have escaped the fagacity of those who planned it; and that it fhould never have occurred that a bill directing, that every contra&i shall be difpofed of by auction to the loweft undertaker, can be the only means of preferving public œconomy in thefe tranfactions; of giving a fair chance to the independent trader, and of preventing effectually the corrupt influence of the minifter. If the contractor does not fulfil the terms of his contract, it will be to his own lofs, as the courts of law will fcarcely be backward in compelling an individual to do juftice to the public. Every neceffary for the fleet and army fhould also be fupplied by contract, and as little left to the rapacity of commiffaries as poffible. The prediction may feem vifionary, but time will probably justify its authors--If ever the liberty of England should be annihilated, it will be by the corrupt influence of administration exerted over the commercial world.'

With refpect to the negociations for the peace in 1782, the author appears to have received good information; and the intriguing fpirit of the old government of France was curioufly difplayed in that tranfaction.

The intriguing fpirit of the French court was completely manifested in the courfe of the negociation. To detach the Americans as fubjects from Great Britain, was the object for which France entered into the war; to detach them finally as allies, was the great

point to be carried in the formation of a treaty of peace. Every effort was made to create a permanent jealoufy between Britain and America; and, ftrange as it may appear, the minifters of France: affected to favour, in the negociation, the claims of the British ministers, rather than thofe of America. The people of Great Britain have been generally tenacious of the right of fishing in the northern quarters of the Atlantic; and the people of the New Eng land states had determined on the full enjoyment of that right. The boundaries which the American commiffioners claimed, were alío objected to by England; and, in both thefe objections, fhe was fupported by France; and the confidential fecretary of the count de Vergennes was dispatched to exhort lord Shelburne to perfift in his refufal. The British minister was not without a confidential friend upon the fpot, a man of uncommon talents, and of the most cook fagacity. By conferring with the American commiffioners, this gentleman was enabled to penetrate the infidious defigns of the French court; he travelled poft to acquaint his friend with the real state of the negociation; and lord Shelburne had the difcernment to fee that conceffions to America in these points were the immediate interest of Great Britain, though not of France; and that the great object of alarm to this latter power was, left America should once more become the friend and ally of her ancient connexion.'

From these extracts our readers will perceive that the hif tory before us is written both with fpirit and elegance, and abounds in forcible and interefting obfervations on the moft important political topics. It was not a little flattering to us to find our judgment on the two laft volumes of Dodiley's Annual Register confirmed by this able and intelligent author; as he has most fatisfactorily detected the compiler of that work in impofing upon a credulous public the groffeft and most ridiculous falfchoods respecting the French Revolution.-But we must defer entering on this part of the work to a future opportunity, having already exceeded our ufual limits.

On the Determination of the Orbits of Comets, according to the Methods of Father Bofcovich and M. De la Place, with new and complete Tables and Examples of the Calculation by both Methods. By Sir Henry Englefield, Bart. 4to. 15s. Boards. Elmfly. 1793

THE

HE difficulties attending the tracing of a comet's path are well-known to every mathematician; and the labours of the most eminent men have been fuccefsfuily employed, either in fhortening the process, or giving greater precifion to this complicated problem. In the work before us are laid down two modes, the one by Bofcovich, taken from his Opufcula, printed at Bafano, in the year 1785; the other by De la

T

I 4

Place,

Place, taken from the Memoires de l'Academie des Sciences for the year 1780.

According to the first mode, from three obfervations of the comet, at an interval of from five to ten days, we have the longitudes, latitudes, and places in the ecliptic of the comet, which is fuppofed to move in a parabolic orbit. From thefe data, with the known properties of the parabola, the motion of a body in it, together with the motion of the arch in its orbit and distance from the fun being given, are determined the dimenfions and pofitions of the parabola, the place of the comet in its orbit at a given time, its diftance from the fun and earth, and its heliocentric longitude and latitude. The velocity of the interfection of the radius vector of the chord is affumed to be nearly equable in arches, whofe verfed fines are fmall compared with the radius vector; and from the interfections of the radius vector of the comet and the earth at the fecond obfervation is deduced the ratio of the curtate distances of the comet from the earth. One being affumed as known, the others are given, and hence the diftances of the comet from the fun and the chord of the parabolic are defcribed by the

*comet.

For the various things required in the mode laid down, feparate chapters are affigned. In one is determined the motion of the point of interfection of the radius vector and chord, in another the parabolic chord is compared with the fpace anfwerin to the mean velocity of the earth in the fame time, in another the proportion of the three curtate ditances of the comet from the earth is laid down, and in the fame manner every other requifite for the determination of the orbit is clearly inveftigated. An application is made of the principies to the comct of 1769, and from thence the reader will be enabled to apply with cafe this mode to any future comet.

From the Memoires de l' Academie des Sciences, our author has taken only one part of De la Place's Memoir. It confifts of two parts, the firft containing the principles on which the method is founded; the fecond, the practical part. The latter part only is given in this work; and we cannot but lament that a work fo eminently useful fhould have omitted fo important a part of the memoir. De la Place's mode is feientific; and the profoundet mathematician might not blush to require an explanation of feveral of its parts. We are working now in the dark, we have our algebraical expreffions, and we have nothing to do but to apply them. Sull every one wifhes to be convinced that the grounds of these rules are good; and as he may not cafily procure the works in which they are laid down, the fame motives which firit occafioned the publication of this work, might have led the author to make it perfect, by giving us the principles as well as the

prac

practice. But as we are encouraged to hope that a hiftory of the principal comets, with much curious matter from scarce books and manufcripts, may hereafter be given as an addition to this work, the author will perhaps fee the propriety of fupplying the deficiency in the part already published, by allotting a place in his future volume to the firft part of De la Place's Memoir.

De la Place's method confifts of two parts; the first determining the approximate perihelion diftance and time of the comet's arrival at the perihelion; the fecond containing the correction of the approximate perihelion diftance and time of perihelion from more diftant obfervations, and a determination of the remaining elements of the comet's orbit.

For the first part, three, four, or five obfervations of the comet are chofen, as nearly equidistant from each other as poffible: the more obfervations are ufed, the greater may be the arc of the comet's motion, and by these means the influence of the errors of obfervation on the operation will be diminished. An epoch is fixed on, being a time equidiftant, or nearly fo, from the two extreme observations; and, by a formula involving the longitudes and differences of the longitudes at the refpective obfervations, and the number of days the epoch is diftant from each obfervation, the longitude of the comet for a time diftant from the epoch is given. By fubftituting the latitudes at each obfervation, inftead of the longitudes in the above formula, it gives the latitude at that time. From the longitude and latitude thus found, the longitude of the earth and its radius vector, and, at the fame time, radius vector for a longitude ninety degrees forwarder in the ecliptic than its place at the epoch, four equations are given, from whence are obtained the value of the curtate dittance of the comet from the earth, the ratio of the elements of distance to the clements of time, and the radius vector of the comet: and having obtained the values of the three laft quantities, the perihelion diftance is found from two equations.

From this approximation the fecond step is to find the exact elements of the orbit; and, for this purpofe, three diftant obfervations are chofen, from which, by the perihelion distance and time of arrival at it already found, the three true anomalies of the comet, and the three radii vectores are computed according to the method given in the ufe of the general table of the parabola. Then from the three obferved geocentric latitudes of the comet, its geocentric longitudes, its elongations, the three correfponding longitudes of the earth and its radii vectores, the heliocentric longitudes and latitudes of the comet are found, and an expreffion is given for the angle at the fim fought, which, if the perihelion distance and time of arrival determined before were exact, would be equal to the

angle

angle found, by fubtracting the first anomaly from the fecond. But as this is fcarcely to be expected, the perihelion distance must be changed, fuppofe a fiftieth, whilft the time of paffing it remains unaltered: and computing as before upon this new hypothefis, the anomalies, radii vectores and angles between them, the errors in the angle at the fun are found. A third hypothefis is now to be formed, in which the perihelion diftance, is fixed according to the firft hypothefis, and the time of paffage at the perihelion is changed a little, fuppose balf a day, or a day, according to the quantity of the errors; and by a calculation fimilar to the foregoing, the error in the angle fought at the fun is determined. From thefe errors thus found, by means of two equations, the true perihelion diftance and time of arrival at the perihelion are obtained.

An inftance is given of the above method in the comet of 1769, for which are first found the approximate perihelion distance and time of arrival at the perihelion, and then the erTors are corrected, and the true elements of the orbit are found.

At the end of the volume are given four tables, the firft, for converting time into decimals of a day; the fecond, for converting decimals of a day into time; the third, is a general table of the parabola, by Delambre; the fourth, is a general table of the parabola by Barker.

From the sketch above given, and the names of Bofcovich and De la Place, the work might recommend itself to every mathematician; and, we may add, that from the great pains taken in fuperintending the publication, it acquires an additional value. The two modes were hid in volumes to which few could have accefs, and we are indebted to the author for laying them before the public in fuch a manner, that all who. are employed in fpeculations of this fort must be highly gratified. We fhall therefore hope that he will continue his useful labours, and, in giving us his hiftory of the comets, take away from us every reason to lament that the Cometographie of Mr. Perigré has not appeared in an English drefs. A work of this kind is very much wanted, and is of great importance to science; for whilst we admire the wisdom and power of the Supreme Being, from our more accurate knowledge of the planetary fyftem, how much muft thofe ideas be enlarged, when we contemplate the vast number of bodies revolving around the fun, the inclinations of whofe orbits are fo various, whofe perihelia are at fuch different diftances, and whose times of revolution from one exceeded by a planet's are at least of such length, that we might almoft apprehend them to be vifitors from other spheres. We have as yet an account, that may be depended on, of very few; for what are feventy-two to the whole number of bodies, which escaping mortal ken, are per