may expofe us to the philofophical excommunication which it contains: But before I take my leave of Plotinus, I cannot refrain from addreffing a few words to the Platonical part of my readers. If fuch, then, is the wisdom contained in the works of this philofopher, as we may conclude from the prefent fpecimen, is it fit fo divine a treafure fhould be concealed in fhameful oblivion? With respect to true philofophy, you must be fenfible, that all modern fects are in a state of barbarous ignorance: for Materialism, and its attendant Senfuality, have darkened the eyes of the many, with the mists of error; and are continually ftrengthening their corporeal tie. And can any thing more effectually diffipate this increasing gloom than difcourfes composed by so fublime a genius, pregnant with the most profound conceptions, and every where full of intellectual light? Can any thing fo thoroughly deftroy the phantom of falfe enthufiafm, as eftablishing the real object of the true? Let us then boldly enlift ourselves under the banners of Plotinus, and, by his affiftance, vigorously repel the encroachments of error, plunge her dominions into the abyss of forgetfulness, and difperfe the darkness of her baneful night. For, indeed, there never was a period which required fo much philofophic exertion, or fuch vehement contention from the lovers of Truth. On all fides, nothing of philofophy remains but the name, and this is become the fubject of the vileft proftitution: fince it is not only engroffed by the Naturalift, Chemift, and Anatomift, but is ufurped by the Mechanic, in every trifling invention, and made fubfervient to the lucre of traffic and merchandize. There cannot furely be a greater proof of the degeneracy of the times than fo unparralleled a degradation, and fo barbarous a perverfion of terms. For, the word philofophy, which implies the love of wisdom, is now become the ornament of folly. In the times of its inventor, and for many fucceeding ages, it was expreffive of modefty and worth; in our days, it is the badge of impudence and vain pretenfions. It was formerly the fymbol of the profound and contemplative genius; it is now the mark of the fuperficial and unthinking practitioner. It was once reverenced by Kings, and clothed in the robes of nobility; it is now (according to its true acceptation) abandoned and defpifed, and ri diculed by the vileft Plebeian. Permit me, then, my friends, to address you in the words of Achilles to Hector: Roufe, then, your forces, this important hour, Collect your strength, and call forth all your pow'r. Since, to adopt the animated language of Neptune to the Greeks, on daftards, dead to fame, I waste no anger, for they feel no shame; But you, the pride, the flower of all our hoft, Nor deem the exhortation impertinent, and the danger groundless Hark! the gates burst, the brazen barriers roar. Impetuous ignorance is thundering at the bulwarks of philofophy, and her facred retreats are in danger of being demolished, through bur feeble refiftance. Rife, then, my friends, and the victory will be be ours. The foe is indeed numerous, but, at the fame time, feeble: and the weapons of truth, in the hands of vigorous union, defcend with irrefiftible force, and are fatal wherever they fall.' ART. VII. Mathematical Effays on several Subjects: containing new Improvements and Discoveries. By the Rev. John Hellins. 4to. 7s. 6d. fewed. Davis. 1788. ་ THE Public are here prefented with a collection of Effays, Twritten, the author fays, to amufe folitude, and with the hopes of producing fomething which might be useful to the community. That the firft intention is fully answered there is no room to doubt, because no amusement can furnish more ra◄ tional pleasure, or afford greater fatisfaction to a contemplative mind, than fpeculative mathematics. But this pleasure can only be known by those who have tasted it; and perhaps it may, like a precious jewel, be held in greater eftimation by its poffeffors, because of the difficulty of obtaining it. Nor is there more reason to doubt that the fecond is alfo fulfilled, because the diffufion of knowlege muft, in the end, become beneficial to the Public. Effay I. On the computation of Logarithms, was published in the 70th volume of the Philofophical Trapfactions; for an account of which, fee Review, vol. lxiv. p. 440. The fecond is on the fame fubject, and contains a variety of theorems for computing logarithms, with a method of conftructing a table of Briggs's logarithms. Had fuch theorems been wanting, and had no tables been already in the hands of the Public, this effay would have been of great ufe to the mathematical world; but although its extenfive utility is not very apparent, yet it difplays great ingenuity and invention, and may afford confiderable affiftance to thofe who are now engaged in making logarithmic tables. The investigation of the theorems here given is too long for infertion in this article, but we are perfuaded that it will afford pleafure to all lovers of algebra; and the theorems themselves, without the investigation, would perhaps be unfatisfactory to the mathematical reader. With refpect to Mr. Hellins's method of conftructing a table of Briggs's logarithms to fourteen places of decimals, it begins with finding the hyperbolical or Napier's logarithm of ten, and thence the modulus 0.434&c. The firft, fecond, third, &c. differences of the larger logarithms are computed from the preceding theorems for that purpose; and the whole table is filled up by different means, which, were we to enumerate them separately, would increase the fize of this article beyond our narrow limits. Effay III. and IV. are On the reduction of Equations that have two equal Roots. One of these effays was published in the Philo fophical fophical Transactions for 1782, an account of which was given in the 69th volume of our Review, p. 457; where the merits of this. paper were particularly examined. The other, which is now first published, contains, befide theorems for finding the equal roots, feveral useful theorems for difcovering whether a given equation has two equal roots. At the end of thefe Effays, the Author fays, that they are two fections of a new fyftem of algebra, planned and begun feveral years ago, in which he intended to treat diftin&ly of equations that have two, as well as those that have three equal roots; and to apply thofe equations to fuch ufes as he has not been able to find in any other book. That they may be applied to feveral uses, which have never yet been mentioned, there is not the leaft doubt; for, in practice, cafes continually occur which require various modes of treatment; yet it must be acknowleged, that equations, with equal roots, are uncommon, and the method of finding the equal roots is often, especially in high equations, a laborious operation, independent of the trouble of determining whether the given equation have equal roots, or not. Effay V. is On the Correction of Fluents found by Defcending Series. The introductory fentence to this Effay is as follows; "Although the finding of fluents by defcending feries has been often mentioned by the writers on fluxions, yet that method does not appear to have been brought into use in the solution of problems, even by thofe late and celebrated writers, Emerfon and Simpfon, who, in their treatises of fluxions, have given no inftances of the actual ufe of fuch feries.' When we first read this paragraph, we thought we had recollected to have seen the defcending feries used for finding fluents; and turning to the 29th and 30th examples to Prop. 1oth of the 1ft fection of Emerfon's Fluxions, we met, in each, with two methods for finding the fluents of a Auxionary equation; one by the afcending, and another by the defcending feries; and the 27th example is exprefsly given for finding the fluent by a defcending feries. ་ Mr. Hellins obferves that the values of the fluents given by the afcending and defcending feries are not equal; that Sir Ifaac Newton had mentioned this difference, but that he had not noticed it to be conftant; and that Mr. Euler had obferved the difference to be conftant, but that the method, which he used for determining it, would not give the quantity fought very accurately.' Thefe particulars being premifed, Mr. Hellins proceeds to the folution of fome problems, in which the difference here mentioned is pointed out, and computed. His firft cafe is to find, the correct fluent of √a+x, where x and begin together. To illuftrate his pofition, he gives the value of in Anite terms, a + x. √e+x-java; in an afcending Rev. Aug. 1788. . feries, L a3 a2 and in a descending feries, =√xX:}x+a+ 4.5 4.6.x &c. He then fhews that the correction a√a, to be applied to the fluent in finite terms, found by the ufual method, is the difference between the two feries:that the afcending feries gives the true fuent; and that the defcending feries is too great by that conftant difference. The method of finding the correction is also given; but it is somewhat laborious; in cafes, however, where the afcending feries diverges, or converges very flowły, this method of computing the value of the fluent, by the defcending feries, is the only one that can be advantageously ufed, and, on that account, is highly valuable. A typographical error occurs near the end of this effay, at p. 112, line 4, viz. Effay VI. On the Transformation of certain Series to others of fwifter Convergency, contains fome curious inventions, of which a calculator will avail himself with advantage. The feries to which Mr. Hellins has applied this method of transformation are those usually employed for computing hyperbolical logarithms and circular ares. It will indeed apply to others, and it may also be infinitely varied. The laft Effay is an inveftigation of the Force of ofcillating Bodies on their Centers of Sufpenfion. The propofitions here given do not admit of any abridgment. They are purely geometrical, and cannot fail of pleafing the mathematical mechanic, by whom indeed they can only be understood. It appears from this enumeration of the contents of Mr. Hellins's volume of Effays, that he has applied himself to some of the more abftrufe parts of mathematics; and that his proficiency in these ftudies is by no means inconfiderable. The tracts of which we have just given an account are all of them useful, and we do not doubt their being well received by all who are judges of their merit. We fhall wait with impatience the publication of a fecond volume, in which we are promifed fome new theorems for extracting the fquare and cube roots, a method of finding products and quotients to eleven or twelve places of figures, by meams of logarithmic tables only to feven places; and feveral improvements in algebra and Auxions. ART. ART. VIII. Memoir of a Map of the Countries comprehended be tween the Black Sea and the Cafpian; with an Account of the Caucafian Nations, and Vocabularies of their Languages. By G. Ellis, Efq. 4to. 9s. Boards. Edwards. 1788. CAT ATHARINE the Second of Ruffia, whofe encouragement of the arts and fciences has been great, and whofe aggrandifement of her empire has been rapid, having completed the lines extending from the Cafpian Sea to that of Afoph, Lat. 43° 45′ (which lines were thrown up as barriers against the incurfions of her barbarian neighbours, the inhabitants of the country known by the general name of Circaffia), fent, as we are informed by our Author, a gentleman of the name of Guldenftaedt, to Mount Caucafus, with orders to traverse those wild regions, in various directions, to trace the rivers to their fources, to take aftronomical obfervations, to examine the natural hiftory of the country, and to collect vocabularies of all the dialects that he might meet with; which might be afterward referred to their refpective languages, fo as to form a general claffification of all the nations comprehended between the Euxine and Caspian. It is hoped,' fays Mr. Ellis in his preface, that the map now offered to the Public, will be found to be much fuller and more accurate than any which has yet been published. It is ftill, however, very imperfect; and many errors will doubtless be difcovered in it, when the countries that it reprefents fhall have been completely and accurately furveyed. To fuch a map it seemed neceffary to annex a few pages of narration, and I flatter myself that I fhall not be thought to have trefpaffed too much on the reader's patience. What I have offered is principally drawn from the first volume of Mr. Guldenftaedt's Travels,-from various papers inferted in the St. Peterf burgh Journal-from Dr. Reineg's Defcription of Georgia, published in a periodical work by Profeffor Pallas-from the materials contained in Muller's Samlang Ruffifche Gefchite, and from fome manufcript relations which it is needless to particularize.' With respect to the accuracy with which the feveral places are laid down in the prefent large and fplendid map, it cannot be expected that we should hazard any opinion. The country is little known to Europeans, the Ruffians excepted; and from ancient writers, nothing fatisfactory is to be gathered concerning its geography, in any of its parts. Were we, however, to judge from the fulness of Mr. Ellis's draught, we fhould imagine (the fituation of the towns, &c. admitted as right) that scarcely any thing remained to be done. But leaving this matter to be determined by the researches of geographers, we proceed to confider that part of the prefent Memoir which brings us acquainted with the people occupying this particular tract of land; i. e. the feveral provinces lying be L 2 tween |