plied to the Determination of the Parallax of Mars. By M. DioNIS DU SEJOUR.

IX. X. XI. XII. XIII. On the total Eclipfe of the Moon, the 18th of March, 1783.-Occultations of fome Stars of the Pleiades, obferved at Paris, February 9th 1783, and compared with the correfpond nt Occultations, cbferved at Bude, Drontheim, and Bagdai.— On the Comet of 1783. By M. MECHAIN.

MATHEMATICS and MECHANICS.

Mem. I. Spherical Trigonometry, deduced, in a compendious and complete Manner, from the algebraical Solution of the most simple of its general Problems, by means of the different Transformations, of which, the Proportions of Sines and Co-fines, of Tangents and Cotangents, Secants and Co fecants of the jame Arch, or the fame plane Angle, render that Solution fufceptible, and containing fome new and ufeful Formule and Obfervations. By the Abbé GUA.

11. Different Measures of spherical Areas and folid Angles, triangular and polygonal, of which a fufficient Number of the Elements is fuppofed to be known. By the Same.

III. New Propofitions not less useful than curious, relative to the Tetraedron, or an Effay on Tetraedrametry. By the Same. IV. Concerning the Approximations of Formula, which are Functions of great Numbers. By M. DE LA PLACE.

V. Concerning the Calculation of Probabilities. Part IV. containing Reflections on the Method of determining the Probability of future, by the Obfervation of past Events. By the Marquis DE CONDORCET.

VI. Theorem on Equations in finite Differences. CHARLES.

By M. VII. Remarks on the Manner of integrating, by Approximation, differential Equations, and Equations of partial Differences. By M. COUSIN.

VIII. Remarks on the Mathematical Theory of the Motion of Fluids. By the Same.

IX. Concerning Births, Marriages, and Deaths at Paris, from the Year 1771 to 1784, and through the whole Extent of France, during the Years 178: and 1782. By M. DE LA PLACE. This Memoir contains an ingenious method of calculating the popu→ lation of a country. There are annexed to it Two Tables, furnifhed by a learned magiftrate, who has been much occupied in this fubject, with a view to public utility. From the firft, which takes in a series of fourteen years, and regards only the city of Paris, it appears that the total lift, during that period, of births, marriages, and deaths, was as follows:

The Second Table, which regards the population of the whole kingdom of France, the ifland of Corfica included, exhibits, for the years 1781 and 1782, the following refults: For 1781, 970,406 Births male and female; 236,503 Marriages; 881,138 Deaths.-For 1782, 975,703 Births; 224,890 Marriages; 948,502 Deaths. The rules which the Academician lays down, to eftimate, from fuch materials as thefe, the population of a country, are learned and ingenious. There are many contingent circumftances, that render it impoffible to come to a rigorous accuracy in calculations of this kind, and thefe M. de la Place points out with his ufual penetration. He fhews, however, by what method of calculation a probability of a thousand to one may be obtained, that an eftimate of the population of the kingdom of France fhall not be inaccurate to the amount of half a million.

Mem. X. Concerning a Method of integrating ordinary differential Equations, when they are of the higher Degrees, and in those Cafes, in which their complete Integrals are Algebraical. By M. MONGE

XI. Concerning the Integration of Equations of finite Differences, which are not linear. By the Same.

ART. III.

The Norwegian Account of Haco's Expedition against Scotland, A. D. 1263, now first published in the original Iflandic from the Flateyan and Frifian Manuscripts, with a literal English Version and Notes. By the Rev. James Johnftone, A. M. Chaplain to his Britannic Majefty's Envoy at the Court of Denmark. 8vo. 3s. Copenhagen printed, 1782, and fold by Cadell, in London. 1786.

THE

HE publication of manuscripts tending to illuftrate the obfcure parts of hiftory, is a work which Mr. Johnftone has been encouraged to undertake, from the peculiar advantages he enjoyed in having accefs to a number of inedited fragments of northern hiftorians. It was his intention to have given a fuccinct detail of the descents made by the northern nations on the British ifles; but a vaft increase of materials obliged him to defer that work to a future period; in the mean time, he favours the antiquary and the hiftorian with fuch detached parts as he hath been able to collect.

The

The fituation of the kings of the ifles was peculiarly delicate; for though their territories were extenfive, yet they were by no means a match for the neighbouring ftates. On this account, allegiance was extorted from them by different fovereigns. The Hebridian princes confidered this involuntary homage as implying protection; and when this was not granted to them, they were under the neceffity of forming fuch new connexions as they fuppofed, or hoped, would be more conducive to their fafety and protection.

The Norwegians were in poffeffion of the Hebrides when the Alexanders of Scotland, after having added Galloway (then a powerful maritime ftate) to their dominions, attempted to expel the Norwegians from the Hebrides. In order to effect their defigns, they firft fecured the Somerlid family, and gained over the infular chiefs. Haco, the Norwegian monarch, was no lefs anxious to attach every perfon of confequence to his party all his pacific efforts were however ineffectual. Difappointed in his negotiations, he had recourfe to the fword; and failing from Norway with a very numerous armament, he defeated the Scots in many fkirmishes, and died at Kirkwall in the winter following.

The publication before us contains a minute detail of this expedition, according to the Flateyan and Frifian MSS. the first of which belongs to the library of his Danifh majefty, and the other to the Magnæan collection. Of these, Mr. Johnstone obtained copies. By the help of one he was enabled to fupply, reciprocally, the imperfections of the other; and to correct many errors of tranfcribers. Thefe MSS. have alfo furnifhed him with several emendations of a former publication, The anecdotes of Olave the Black*', which corrections he has inferted in the preface to this performance.

As antiquaries may be defirous of knowing fomething of these MSS. Mr. Johnftone has given the following defcription of them:

The Frifian MS. is a vellum Quarto of the largeft fize in a beautiful hand, and the character resembles that which prevailed in the end of the thirteenth century. The book of Flatey is a very large vellum volume in folio, and appears to have been compiled in the fourteenth age. It contains a collection of poems; excerpts from Adam Bremenfis; a differtation on the firft inhabitants of Norway; the life of Eric the traveller; Olave Trygvafon; of St. Olave; of the Earls of Orkney; of Suerir; of Hace the aged; of his fon Magnus; of Magnus the Good; of Harald the Imperious; of EinaSockafon of Greenland; and of ölver the Mifchievous; it contains alfo a general chronology down to A. D. 1394, the year in which the MS. was completed. The initial letters, in fome places, are or

* See Review, vol. LXV. p. 95.
૨.g

namented

namented with hiftorical miniature paintings, in which the figures are moftly in armour or mail; their helmets are fometimes conical, fometimes like a broad-brimmed hat.'

The prefent publication will be an acceptable prefent to the hiftorian and antiquary, while the philological Critic will be entertained with the fpecimens of Islandic poetry which are introduced as explanatory of the text, or in confirmation of the facts. When Haco fet fail, he is thus defcribed by Sturla in the Ravens ode:

*

Leyfti laug raftar
Landa ftyrandi
Helldo há falldar

Húfom blá dúfor.

Lyftiz hrein heftom
Höfn af fkip ftöfnom
Elldi álfolidar

Authr glód raudom.

In English The leader of his people unmoored the ploughers of the ocean. He raised aloft the expanded wings f of the fky-blue dovest. Our fovereign, rich in the fpoils || of the fea-inake's den, viewed the retiring haven from the ftern of his fnorting Reed ¶ adorned with ruddy gold.'

The narrative itfelf is in a plain and rather uncouth ftyle; the tranflation is clofe, and in fome places fo literal, as to occafion fome obfcurity which may be unpleafant to an English Reader; it conveys, however, the fenfe of the original; and is a good reprefentation of the great fimplicity of the language of Re

these times.

AR T. IV.

Antiquitates Celto-Normannica, containing the Chronicle of Man and the Ifles, abridged by Camden, and now first published, complete, from the original MS. in the British Mufeum; with an English Tranflation and Notes, &c. By the Rev. James Johnstone, A. M. Rector of Maghera Crofs, and Member of the Royal Societies of Edinburgh and Copenhagen, 4to. 10s. 6d. Printed at Copenhagen, and fold by Cadell. London. 1786.

N publifhing a work of this kind it is ufual, and even highly neceffary, that the Editor fhould not only inform his Readers whence the materials are taken, but defcribe the manufcripts which contain them. Though there is not the leaft reafon to doubt, that Mr. Johnftore, from his knowledge of the northern antiquities and languages, would publifh any thing, the authen

+ Sails.

Ship.

Ships.

ShipsMythologifts frequently reprefent treafures as guarded by dragons, &c.

ticity of whofe originals were dubious; yet fuch is the curiofity of the antiquary, that a prefatory introduction, explaining and defcribing the MSS. would be fatisfactory and acceptable; Mr. Johnstone has however omitted giving this intelligence, which appears to be the more neceffary, fince the dates of fome material tranfactions recorded in this work, differ widely from thofe in common ufe, which have ever been esteemed jun.

The publication before us contains fix diftinct works. The firft is intitled Chronicon Manniæ it Infularum. This Chronicle begins with the year of Chrift 1000, which the Editor has altered to 1015, without affigning any other reafon than what he fays in the following note: It is neceffary to premife, that, in the original MS. feveral of the dates have evidently been erafed, and others inferted by fome ignorant tranfcriber.' Mr. Camden, in his Britannia, has inferted in the Appendix, A Chronicle of the Kings of Man, which commences with the year 1065, and ends, as Mr. Johntone's, with 1316; but the intermediate dates differ, though the facts recorded in both are the fame. Camden, in his abridgment, has given the tranflation only, while Mr. Johnstone gives the original, with his Tranflation in oppofite columns. The two tranflations differ in fome respect; Mr. Johnstone's feems, on the whole, more accurate; and fhews, in fome inftances, his thorough acquaintance with the hiftory of the times. One paffage, however, is fo differently tranflated, that we cannot but notice it: Scotos ita perdomuit, ut nullus qui navem vel fcapham fabricavit, aufus effet plus quam tres clavos inferere. He humbled the Scotch to fuch a degree, that no hip-builder durft ufe above three bolts in any veffel.' Johnstone. He brought the Scotch to fuch fubje&tion, that if any of them built a fhip or a boat, they were not allowed to have above three ferns in it.' Camden. If we fuppofe the writer of this Chronicle, who was probably a monk of the 13th century, to have been a good Latinift, he might have ufed the word clavus, which literally fignifies a bolt or beam, for the helm or ftern of a ship, as Cicero does in his treatife on Old Age, chap. 6. where he fays, Ille [fc. gubernator] clavum tenens fedet in puppi.' The paffage would perhaps be better understood, had we a perfect knowledge of the construction of the fhips ufed in those days.

The fecond work is an extract from the annals of Úlfter, not taken from the original, but from a verfion partly Latin and partly English, in the British Mufeum. The Editor's principal reafon for printing these Extracts, was, the hope that such a fpecimen might fuggeft to fome Irish gentleman the idea of pubblifhing thefe valuable records in the original. This work contains the principal tranfactions in Ireland, Scotland, and other northern nations, from the year 431 to 1302.

The